awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
⊢ (nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
      { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } =
    a

Tactic:
case hom_id => exact ((h a).mp ha).2.nonUnitalStarAlgHom_id <| cfcₙHom_id ((h a).mp ha).1

Premises used: QuasispectrumRestricts.nonUnitalStarAlgHom_id, cfcₙHom_id, And.right, And.left, Iff.mp
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
simp :
  ∀ (f_1 : C((↑(σₙ R a) : Type u_1), R)₀) (x : R),
    (∃ (y : (↑(σₙ S a) : Type u_2)),
        (f_1 : (a : (↑(σₙ R a) : Type u_1)) → R)
            (Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯ y) =
          x) ↔
      ∃ (y : (↑(σₙ R a) : Type u_1)), (f_1 : (a : (↑(σₙ R a) : Type u_1)) → R) y = x
⊢ ∀ (f_1 : C((↑(σₙ R a) : Type u_1), R)₀),
    quasispectrum.{u_1, u_3} R (A := A)
        ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) f_1 : A) =
      range (ι := (↑(σₙ R a) : Type u_1)) (⇑f_1 : (a : (↑(σₙ R a) : Type u_1)) → R)

Tactic:
simp only [nonUnitalStarAlgHom_apply, ← @quasispectrum.preimage_algebraMap (R := R) S,
        cfcₙHom_map_quasispectrum, Set.ext_iff, Set.mem_preimage, Set.mem_range, comp_apply, coe_mk,
        ContinuousMap.coe_mk, StarAlgHom.ofId_apply, halg.injective.eq_iff]

Premises used: Set.mem_range, Function.Injective.eq_iff, ContinuousMapZero.coe_mk, StarAlgHom.ofId_apply, Topology.IsClosedEmbedding.toIsEmbedding, cfcₙHom_map_quasispectrum, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, ContinuousMap.coe_mk, Topology.IsEmbedding.injective, Set.ext_iff, ContinuousMapZero.comp_apply, quasispectrum.preimage_algebraMap, Set.mem_preimage
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
⊢ ∀ (f_1 : C((↑(σₙ R a) : Type u_1), R)₀) (x : R),
    (∃ (y : (↑(σₙ S a) : Type u_2)),
        (f_1 : (a : (↑(σₙ R a) : Type u_1)) → R)
            (Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯ y) =
          x) ↔
      ∃ (y : (↑(σₙ R a) : Type u_1)), (f_1 : (a : (↑(σₙ R a) : Type u_1)) → R) y = x

Tactic:
fun _ _ ↦ ((h a).mp ha).2.homeomorph.exists_congr fun b ↦ Iff.rfl

Premises used: Homeomorph.toEquiv, Iff.rfl, And.right, Iff.mp, QuasispectrumRestricts.homeomorph, Equiv.exists_congr
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
⊢ ∀ (f_1 : C((↑(σₙ R a) : Type u_1), R)₀) (x : R),
    (∃ (y : (↑(σₙ S a) : Type u_2)),
        (f_1 : (a : (↑(σₙ R a) : Type u_1)) → R)
            (Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯ y) =
          x) ↔
      ∃ (y : (↑(σₙ R a) : Type u_1)), (f_1 : (a : (↑(σₙ R a) : Type u_1)) → R) y = x

Tactic:
exact fun _ _ ↦ ((h a).mp ha).2.homeomorph.exists_congr fun b ↦ Iff.rfl

Premises used: Homeomorph.toEquiv, Iff.rfl, And.right, Iff.mp, QuasispectrumRestricts.homeomorph, Equiv.exists_congr
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
⊢ ∀ (f_1 : C((↑(σₙ R a) : Type u_1), R)₀),
    quasispectrum.{u_1, u_3} R (A := A)
        ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) f_1 : A) =
      range (ι := (↑(σₙ R a) : Type u_1)) (⇑f_1 : (a : (↑(σₙ R a) : Type u_1)) → R)

Tactic:
case hom_map_spectrum =>
      simp only [nonUnitalStarAlgHom_apply, ← @quasispectrum.preimage_algebraMap (R := R) S,
        cfcₙHom_map_quasispectrum, Set.ext_iff, Set.mem_preimage, Set.mem_range, comp_apply, coe_mk,
        ContinuousMap.coe_mk, StarAlgHom.ofId_apply, halg.injective.eq_iff]
      exact fun _ _ ↦ ((h a).mp ha).2.homeomorph.exists_congr fun b ↦ Iff.rfl

Premises used: Set.mem_range, Homeomorph.toEquiv, Iff.rfl, Function.Injective.eq_iff, ContinuousMapZero.coe_mk, StarAlgHom.ofId_apply, Topology.IsClosedEmbedding.toIsEmbedding, cfcₙHom_map_quasispectrum, And.right, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, ContinuousMap.coe_mk, Topology.IsEmbedding.injective, Iff.mp, Set.ext_iff, QuasispectrumRestricts.homeomorph, ContinuousMapZero.comp_apply, quasispectrum.preimage_algebraMap, Set.mem_preimage, Equiv.exists_congr
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
g : C((↑(σₙ R a) : Type u_1), R)₀
rw :
  q ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g) ∧
    QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) (A := A)
      ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A) (⇑f : (a : S) → R)
⊢ p ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g)

Tactic:
rw [h]
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
g : C((↑(σₙ R a) : Type u_1), R)₀
⊢ q ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g) ∧
    QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) (A := A)
      ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A) (⇑f : (a : S) → R)

Tactic:
⟨cfcₙHom_predicate _ _, ?_⟩

Premises used: cfcₙHom_predicate, And.intro
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
g : C((↑(σₙ R a) : Type u_1), R)₀
refine :
  QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) (A := A)
    ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A) (⇑f : (a : S) → R)
⊢ q ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g) ∧
    QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) (A := A)
      ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A) (⇑f : (a : S) → R)

Tactic:
refine ⟨cfcₙHom_predicate _ _, ?_⟩
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
g : C((↑(σₙ R a) : Type u_1), R)₀
⊢ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) (A := A)
    ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A) (⇑f : (a : S) → R)

Tactic:
{ rightInvOn := fun s hs ↦ ?_, left_inv := ((h a).mp ha).2.left_inv }

Premises used: QuasispectrumRestricts.mk, And.right, Iff.mp, QuasispectrumRestricts.left_inv
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
g : C((↑(σₙ R a) : Type u_1), R)₀
refine :
  ∀ (s : S)
    (hs :
      s ∈
        quasispectrum.{u_2, u_3} S (A := A)
          ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A)),
    (algebraMap R S : (a : R) → S) ((f : (a : S) → R) s : R) = s
⊢ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) (A := A)
    ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A) (⇑f : (a : S) → R)

Tactic:
refine { rightInvOn := fun s hs ↦ ?_, left_inv := ((h a).mp ha).2.left_inv }
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
g : C((↑(σₙ R a) : Type u_1), R)₀
s : S
hs :
  s ∈
    quasispectrum.{u_2, u_3} S (A := A)
      ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A)
⊢ Membership.mem (γ := Set S)
    (range (ι := (↑(σₙ S a) : Type u_2))
      (⇑(comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ }
            (g.comp
              {
                toFun :=
                  Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯,
                continuous_toFun := ⋯, map_zero' := ⋯ :
                ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) })) :
        (a : (↑(σₙ S a) : Type u_2)) → S))
    s

Tactic:
rw [nonUnitalStarAlgHom_apply,
        cfcₙHom_map_quasispectrum] at hs

Premises used: cfcₙHom_map_quasispectrum, QuasispectrumRestricts.nonUnitalStarAlgHom_apply
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
g : C((↑(σₙ R a) : Type u_1), R)₀
s : S
hs :
  Membership.mem (γ := Set S)
    (range (ι := (↑(σₙ S a) : Type u_2))
      (⇑(comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ }
            (g.comp
              {
                toFun :=
                  Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯,
                continuous_toFun := ⋯, map_zero' := ⋯ :
                ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) })) :
        (a : (↑(σₙ S a) : Type u_2)) → S))
    s
obtain :
  ∀ (r : (↑(σₙ S a) : Type u_2)),
    Eq.{u_2 + 1} (α := S)
      ((algebraMap R S : (a : R) → S)
          ((f : (a : S) → R)
              ((comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ }
                      (g.comp
                        {
                          toFun :=
                            Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a)
                              (⇑f : (a : S) → R) ⋯,
                          continuous_toFun := ⋯, map_zero' := ⋯ :
                          ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) }) :
                    (a : (↑(σₙ S a) : Type u_2)) → S)
                  r :
                S) :
            R) :
        S)
      ((comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ }
              (g.comp
                {
                  toFun :=
                    Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯,
                  continuous_toFun := ⋯, map_zero' := ⋯ :
                  ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) }) :
            (a : (↑(σₙ S a) : Type u_2)) → S)
          r :
        S)
⊢ (algebraMap R S : (a : R) → S) ((f : (a : S) → R) s : R) = s

Tactic:
obtain ⟨r, rfl⟩ := hs

Premises used: rfl
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
g : C((↑(σₙ R a) : Type u_1), R)₀
r : (↑(σₙ S a) : Type u_2)
⊢ Eq.{u_2 + 1} (α := S)
    ((algebraMap R S : (a : R) → S)
        ((f : (a : S) → R)
            ((comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ }
                    (g.comp
                      {
                        toFun :=
                          Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R)
                            ⋯,
                        continuous_toFun := ⋯, map_zero' := ⋯ :
                        ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) }) :
                  (a : (↑(σₙ S a) : Type u_2)) → S)
                r :
              S) :
          R) :
      S)
    ((comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ }
            (g.comp
              {
                toFun :=
                  Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯,
                continuous_toFun := ⋯, map_zero' := ⋯ :
                ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) }) :
          (a : (↑(σₙ S a) : Type u_2)) → S)
        r :
      S)

Tactic:
simp [((h a).mp ha).2.left_inv _]

Premises used: And.right, Iff.mp, QuasispectrumRestricts.left_inv