Theorem: NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply_comp_toReal
File: Mathlib/Topology/ContinuousFunction/UniqueCFC.lean

tactic: simp only [realContinuousMapZeroOfNNReal_apply]
state_before: X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ.realContinuousMapZeroOfNNReal ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f) = φ f
state_after: X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal -
      φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal =
    φ f

tactic: convert_to φ f - φ 0 = φ f using 2
state_before: X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal -
      φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal =
    φ f
state_after: case h.e'_2.h.e'_5
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ f

case h.e'_2.h.e'_6
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ 0

X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ f - φ 0 = φ f

tactic: on_goal -1 => rw [map_zero, sub_zero]
state_before: case h.e'_2.h.e'_5
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ f

case h.e'_2.h.e'_6
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ 0

X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ f - φ 0 = φ f
state_after: case h.e'_2.h.e'_5
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ f

case h.e'_2.h.e'_6
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ 0

tactic: all_goals
  congr
  ext x
  simp
state_before: case h.e'_2.h.e'_5
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ f

case h.e'_2.h.e'_6
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ 0
state_after: no goals

tactic: rw [map_zero, sub_zero]
state_before: X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ f - φ 0 = φ f
state_after: no goals

tactic: congr
state_before: case h.e'_2.h.e'_6
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ 0
state_after: case h.e'_2.h.e'_6.h.e_6.h
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = 0

tactic: ext x
state_before: case h.e'_2.h.e'_6.h.e_6.h
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = 0
state_after: case h.e'_2.h.e'_6.h.e_6.h.h.a
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
x : X
⊢ ↑((-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal x) = ↑(0 x)

tactic: simp
state_before: case h.e'_2.h.e'_6.h.e_6.h.h.a
X : Type u_1
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Zero X
A : Type u_2
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
x : X
⊢ ↑((-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal x) = ↑(0 x)
state_after: no goals