Theorem: cfc_mul
File: Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean

tactic: by_cases ha : p a
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g✝ : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg✝ : autoParam (ContinuousOn g✝ (spectrum R a✝)) _auto✝
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
⊢ cfc (fun x => f x * g x) a = cfc f a * cfc g a
state_after: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g✝ : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg✝ : autoParam (ContinuousOn g✝ (spectrum R a✝)) _auto✝
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : p a
⊢ cfc (fun x => f x * g x) a = cfc f a * cfc g a

case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g✝ : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg✝ : autoParam (ContinuousOn g✝ (spectrum R a✝)) _auto✝
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : ¬p a
⊢ cfc (fun x => f x * g x) a = cfc f a * cfc g a

tactic: rw [cfc_apply f a, cfc_apply g a, ← map_mul, cfc_apply _ a]
state_before: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g✝ : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg✝ : autoParam (ContinuousOn g✝ (spectrum R a✝)) _auto✝
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : p a
⊢ cfc (fun x => f x * g x) a = cfc f a * cfc g a
state_after: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g✝ : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg✝ : autoParam (ContinuousOn g✝ (spectrum R a✝)) _auto✝
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).restrict fun x => f x * g x, continuous_toFun := ⋯ } =
    (cfcHom ha)
      ({ toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } *
        { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ })

tactic: congr
state_before: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g✝ : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg✝ : autoParam (ContinuousOn g✝ (spectrum R a✝)) _auto✝
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).restrict fun x => f x * g x, continuous_toFun := ⋯ } =
    (cfcHom ha)
      ({ toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } *
        { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ })
state_after: no goals

tactic: simp [cfc_apply_of_not_predicate a ha]
state_before: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g✝ : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg✝ : autoParam (ContinuousOn g✝ (spectrum R a✝)) _auto✝
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : ¬p a
⊢ cfc (fun x => f x * g x) a = cfc f a * cfc g a
state_after: no goals