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

tactic: simp [cfc_const r a]
state_before: case h_C
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 : R → R
a : A
ha : autoParam (p a) _auto✝
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
r : R
⊢ cfc (fun x => eval (f x) (C r)) a = (aeval (cfc f a)) (C r)
state_after: no goals

tactic: simp only [eval_mul, eval_C, eval_pow, eval_X, map_mul, aeval_C, map_pow, aeval_X]
state_before: case h_monomial
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✝¹ : A
ha✝ : autoParam (p a✝¹) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝¹)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝¹)) _auto✝
f : R → R
a : A
ha : autoParam (p a) _auto✝
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
n : ℕ
r : R
a✝ : cfc (fun x => eval (f x) (C r * X ^ n)) a = (aeval (cfc f a)) (C r * X ^ n)
⊢ cfc (fun x => eval (f x) (C r * X ^ (n + 1))) a = (aeval (cfc f a)) (C r * X ^ (n + 1))
state_after: case h_monomial
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✝¹ : A
ha✝ : autoParam (p a✝¹) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝¹)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝¹)) _auto✝
f : R → R
a : A
ha : autoParam (p a) _auto✝
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
n : ℕ
r : R
a✝ : cfc (fun x => eval (f x) (C r * X ^ n)) a = (aeval (cfc f a)) (C r * X ^ n)
⊢ cfc (fun x => r * f x ^ (n + 1)) a = (algebraMap R A) r * cfc f a ^ (n + 1)

tactic: rw [cfc_const_mul .., cfc_pow _ (n + 1) _, ← smul_eq_mul, algebraMap_smul]
state_before: case h_monomial
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✝¹ : A
ha✝ : autoParam (p a✝¹) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝¹)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝¹)) _auto✝
f : R → R
a : A
ha : autoParam (p a) _auto✝
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
n : ℕ
r : R
a✝ : cfc (fun x => eval (f x) (C r * X ^ n)) a = (aeval (cfc f a)) (C r * X ^ n)
⊢ cfc (fun x => r * f x ^ (n + 1)) a = (algebraMap R A) r * cfc f a ^ (n + 1)
state_after: no goals