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

tactic: cases n with
| ofNat _ => simpa using cfcUnits_pow f a hf' _
| negSucc n =>
  simp only [zpow_negSucc, ← inv_pow]
  ext
  exact cfc_pow (hf := hf.inv₀ hf') _ |>.symm
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : HasContinuousInv₀ R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f : R → R
a : A
inst✝ : UniqueContinuousFunctionalCalculus R A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
n : ℤ
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
⊢ cfcUnits f a hf' hf ha ^ n = cfcUnits (f ^ n) a ⋯ ⋯ ha
state_after: no goals

tactic: simpa using cfcUnits_pow f a hf' _
state_before: case ofNat
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : HasContinuousInv₀ R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f : R → R
a : A
inst✝ : UniqueContinuousFunctionalCalculus R A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
a✝ : ℕ
⊢ cfcUnits f a hf' hf ha ^ Int.ofNat a✝ = cfcUnits (f ^ Int.ofNat a✝) a ⋯ ⋯ ha
state_after: no goals

tactic: simp only [zpow_negSucc, ← inv_pow]
state_before: case negSucc
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : HasContinuousInv₀ R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f : R → R
a : A
inst✝ : UniqueContinuousFunctionalCalculus R A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
n : ℕ
⊢ cfcUnits f a hf' hf ha ^ Int.negSucc n = cfcUnits (f ^ Int.negSucc n) a ⋯ ⋯ ha
state_after: case negSucc
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : HasContinuousInv₀ R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f : R → R
a : A
inst✝ : UniqueContinuousFunctionalCalculus R A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
n : ℕ
⊢ (cfcUnits f a hf' hf ha)⁻¹ ^ (n + 1) = cfcUnits (f⁻¹ ^ (n + 1)) a ⋯ ⋯ ha

tactic: ext
state_before: case negSucc
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : HasContinuousInv₀ R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f : R → R
a : A
inst✝ : UniqueContinuousFunctionalCalculus R A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
n : ℕ
⊢ (cfcUnits f a hf' hf ha)⁻¹ ^ (n + 1) = cfcUnits (f⁻¹ ^ (n + 1)) a ⋯ ⋯ ha
state_after: case negSucc.a
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : HasContinuousInv₀ R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f : R → R
a : A
inst✝ : UniqueContinuousFunctionalCalculus R A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
n : ℕ
⊢ ↑((cfcUnits f a hf' hf ha)⁻¹ ^ (n + 1)) = ↑(cfcUnits (f⁻¹ ^ (n + 1)) a ⋯ ⋯ ha)

tactic: exact cfc_pow (hf := hf.inv₀ hf') _ |>.symm
state_before: case negSucc.a
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : HasContinuousInv₀ R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f : R → R
a : A
inst✝ : UniqueContinuousFunctionalCalculus R A
hf' : ∀ x ∈ spectrum R a, f x ≠ 0
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
n : ℕ
⊢ ↑((cfcUnits f a hf' hf ha)⁻¹ ^ (n + 1)) = ↑(cfcUnits (f⁻¹ ^ (n + 1)) a ⋯ ⋯ ha)
state_after: no goals