Theorem: polynomialFunctions.comap_compRightAlgHom_iccHomeoI
File: Mathlib/Topology/ContinuousFunction/Polynomial.lean

tactic: ext f
state_before: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
⊢ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) =
    polynomialFunctions (Set.Icc a b)
state_after: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) ↔
    f ∈ polynomialFunctions (Set.Icc a b)

tactic: fconstructor
state_before: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) ↔
    f ∈ polynomialFunctions (Set.Icc a b)
state_after: case h.mp
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) →
    f ∈ polynomialFunctions (Set.Icc a b)

case h.mpr
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ polynomialFunctions (Set.Icc a b) →
    f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I)

tactic: rintro ⟨p, ⟨-, w⟩⟩
state_before: case h.mp
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) →
    f ∈ polynomialFunctions (Set.Icc a b)
state_after: case h.mp.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ↑(toContinuousMapOnAlgHom I) p = ↑(compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) f
⊢ f ∈ polynomialFunctions (Set.Icc a b)

tactic: rw [DFunLike.ext_iff] at w
state_before: case h.mp.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ↑(toContinuousMapOnAlgHom I) p = ↑(compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) f
⊢ f ∈ polynomialFunctions (Set.Icc a b)
state_after: case h.mp.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), (↑(toContinuousMapOnAlgHom I) p) x = (↑(compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) f) x
⊢ f ∈ polynomialFunctions (Set.Icc a b)

tactic: dsimp at w
state_before: case h.mp.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), (↑(toContinuousMapOnAlgHom I) p) x = (↑(compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) f) x
⊢ f ∈ polynomialFunctions (Set.Icc a b)
state_after: case h.mp.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
⊢ f ∈ polynomialFunctions (Set.Icc a b)

tactic: let q := p.comp ((b - a)⁻¹ • Polynomial.X + Polynomial.C (-a * (b - a)⁻¹))
state_before: case h.mp.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
⊢ f ∈ polynomialFunctions (Set.Icc a b)
state_after: case h.mp.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
⊢ f ∈ polynomialFunctions (Set.Icc a b)

tactic: refine ⟨q, ⟨?_, ?_⟩⟩
state_before: case h.mp.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
⊢ f ∈ polynomialFunctions (Set.Icc a b)
state_after: case h.mp.intro.intro.refine_1
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
⊢ q ∈ ↑⊤.toSubsemiring

case h.mp.intro.intro.refine_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
⊢ ↑(toContinuousMapOnAlgHom (Set.Icc a b)) q = f

tactic: simp
state_before: case h.mp.intro.intro.refine_1
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
⊢ q ∈ ↑⊤.toSubsemiring
state_after: no goals

tactic: ext x
state_before: case h.mp.intro.intro.refine_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
⊢ ↑(toContinuousMapOnAlgHom (Set.Icc a b)) q = f
state_after: case h.mp.intro.intro.refine_2.h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ (↑(toContinuousMapOnAlgHom (Set.Icc a b)) q) x = f x

tactic: simp only [q, neg_mul, RingHom.map_neg, RingHom.map_mul, AlgHom.coe_toRingHom,
  Polynomial.eval_X, Polynomial.eval_neg, Polynomial.eval_C, Polynomial.eval_smul,
  smul_eq_mul, Polynomial.eval_mul, Polynomial.eval_add, Polynomial.coe_aeval_eq_eval,
  Polynomial.eval_comp, Polynomial.toContinuousMapOnAlgHom_apply,
  Polynomial.toContinuousMapOn_apply, Polynomial.toContinuousMap_apply]
state_before: case h.mp.intro.intro.refine_2.h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ (↑(toContinuousMapOnAlgHom (Set.Icc a b)) q) x = f x
state_after: case h.mp.intro.intro.refine_2.h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ Polynomial.eval ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) p = f x

tactic: convert w ⟨_, _⟩
state_before: case h.mp.intro.intro.refine_2.h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ Polynomial.eval ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) p = f x
state_after: case h.e'_3.h.e'_6
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ x = (iccHomeoI a b h).symm ⟨(b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹), ?h.mp.intro.intro.refine_2.h.convert_2⟩

case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ (b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹) ∈ I

tactic: ext
state_before: case h.e'_3.h.e'_6
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ x = (iccHomeoI a b h).symm ⟨(b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹), ?h.mp.intro.intro.refine_2.h.convert_2⟩
state_after: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ ↑x = ↑((iccHomeoI a b h).symm ⟨(b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹), ?h.mp.intro.intro.refine_2.h.convert_2⟩)

tactic: simp only [iccHomeoI_symm_apply_coe, Subtype.coe_mk]
state_before: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ ↑x = ↑((iccHomeoI a b h).symm ⟨(b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹), ?h.mp.intro.intro.refine_2.h.convert_2⟩)
state_after: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) + a

tactic: replace h : b - a ≠ 0 := sub_ne_zero_of_ne h.ne.symm
state_before: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) + a
state_after: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h✝ : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h✝).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
h : b - a ≠ 0
⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) + a

tactic: simp only [mul_add]
state_before: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h✝ : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h✝).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
h : b - a ≠ 0
⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹)) + a
state_after: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h✝ : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h✝).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
h : b - a ≠ 0
⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x) + (b - a) * -(a * (b - a)⁻¹) + a

tactic: field_simp
state_before: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h✝ : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h✝).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
h : b - a ≠ 0
⊢ ↑x = (b - a) * ((b - a)⁻¹ * ↑x) + (b - a) * -(a * (b - a)⁻¹) + a
state_after: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h✝ : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h✝).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
h : b - a ≠ 0
⊢ ↑x * (b - a) = ↑x * (b - a) + -((b - a) * a) + a * (b - a)

tactic: ring
state_before: case h.e'_3.h.e'_6.a
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h✝ : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h✝).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
h : b - a ≠ 0
⊢ ↑x * (b - a) = ↑x * (b - a) + -((b - a) * a) + a * (b - a)
state_after: no goals

tactic: rw [mul_comm (b - a)⁻¹, ← neg_mul, ← add_mul, ← sub_eq_add_neg]
state_before: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ (b - a)⁻¹ * ↑x + -(a * (b - a)⁻¹) ∈ I
state_after: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ (↑x - a) * (b - a)⁻¹ ∈ I

tactic: have w₁ : 0 < (b - a)⁻¹ := inv_pos.mpr (sub_pos.mpr h)
state_before: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ (↑x - a) * (b - a)⁻¹ ∈ I
state_after: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
⊢ (↑x - a) * (b - a)⁻¹ ∈ I

tactic: have w₂ : 0 ≤ (x : ℝ) - a := sub_nonneg.mpr x.2.1
state_before: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
⊢ (↑x - a) * (b - a)⁻¹ ∈ I
state_after: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
⊢ (↑x - a) * (b - a)⁻¹ ∈ I

tactic: have w₃ : (x : ℝ) - a ≤ b - a := sub_le_sub_right x.2.2 a
state_before: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
⊢ (↑x - a) * (b - a)⁻¹ ∈ I
state_after: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ (↑x - a) * (b - a)⁻¹ ∈ I

tactic: fconstructor
state_before: case h.mp.intro.intro.refine_2.h.convert_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ (↑x - a) * (b - a)⁻¹ ∈ I
state_after: case h.mp.intro.intro.refine_2.h.convert_2.left
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ 0 ≤ (↑x - a) * (b - a)⁻¹

case h.mp.intro.intro.refine_2.h.convert_2.right
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ (↑x - a) * (b - a)⁻¹ ≤ 1

tactic: exact mul_nonneg w₂ (le_of_lt w₁)
state_before: case h.mp.intro.intro.refine_2.h.convert_2.left
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ 0 ≤ (↑x - a) * (b - a)⁻¹
state_after: no goals

tactic: rw [← div_eq_mul_inv, div_le_one (sub_pos.mpr h)]
state_before: case h.mp.intro.intro.refine_2.h.convert_2.right
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ (↑x - a) * (b - a)⁻¹ ≤ 1
state_after: case h.mp.intro.intro.refine_2.h.convert_2.right
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ ↑x - a ≤ b - a

tactic: exact w₃
state_before: case h.mp.intro.intro.refine_2.h.convert_2.right
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ ↑x - a ≤ b - a
state_after: no goals

tactic: rintro ⟨p, ⟨-, rfl⟩⟩
state_before: case h.mpr
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ polynomialFunctions (Set.Icc a b) →
    f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I)
state_after: case h.mpr.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
⊢ ↑(toContinuousMapOnAlgHom (Set.Icc a b)) p ∈
    Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I)

tactic: let q := p.comp ((b - a) • Polynomial.X + Polynomial.C a)
state_before: case h.mpr.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
⊢ ↑(toContinuousMapOnAlgHom (Set.Icc a b)) p ∈
    Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I)
state_after: case h.mpr.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
⊢ ↑(toContinuousMapOnAlgHom (Set.Icc a b)) p ∈
    Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I)

tactic: refine ⟨q, ⟨?_, ?_⟩⟩
state_before: case h.mpr.intro.intro
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
⊢ ↑(toContinuousMapOnAlgHom (Set.Icc a b)) p ∈
    Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I)
state_after: case h.mpr.intro.intro.refine_1
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
⊢ q ∈ ↑⊤.toSubsemiring

case h.mpr.intro.intro.refine_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
⊢ ↑(toContinuousMapOnAlgHom I) q =
    ↑(compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (↑(toContinuousMapOnAlgHom (Set.Icc a b)) p)

tactic: simp
state_before: case h.mpr.intro.intro.refine_1
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
⊢ q ∈ ↑⊤.toSubsemiring
state_after: no goals

tactic: ext x
state_before: case h.mpr.intro.intro.refine_2
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
⊢ ↑(toContinuousMapOnAlgHom I) q =
    ↑(compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (↑(toContinuousMapOnAlgHom (Set.Icc a b)) p)
state_after: case h.mpr.intro.intro.refine_2.h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
x : ↑I
⊢ (↑(toContinuousMapOnAlgHom I) q) x =
    (↑(compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (↑(toContinuousMapOnAlgHom (Set.Icc a b)) p)) x

tactic: simp [q, mul_comm]
state_before: case h.mpr.intro.intro.refine_2.h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
x : ↑I
⊢ (↑(toContinuousMapOnAlgHom I) q) x =
    (↑(compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (↑(toContinuousMapOnAlgHom (Set.Icc a b)) p)) x
state_after: no goals