Theorem: StarAlgHom.realContinuousMapOfNNReal_apply_comp_toReal
File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unique.lean

Proof steps:
state_before: X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ.realContinuousMapOfNNReal ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f) = φ f
state_after: X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal -
      φ (-{ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal =
    φ f
tactic: simp only [realContinuousMapOfNNReal_apply]
state_before: X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal -
      φ (-{ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal =
    φ f
state_after: case h.e'_2.h.e'_5
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ f

case h.e'_2.h.e'_6
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ 0

X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ f - φ 0 = φ f
tactic: convert_to φ f - φ 0 = φ f using 2
state_before: case h.e'_2.h.e'_5
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ f

case h.e'_2.h.e'_6
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ 0

X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ f - φ 0 = φ f
state_after: case h.e'_2.h.e'_5
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ f

case h.e'_2.h.e'_6
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ 0
tactic: on_goal -1 => rw [map_zero, sub_zero]
state_before: case h.e'_2.h.e'_5
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ f

case h.e'_2.h.e'_6
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℝ A
inst✝ : TopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ 0
state_after: no goals
tactic: all_goals
    congr
    ext x
    simp