-- Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean
lemma cfc_le_one (f : R → R) (a : A) (h : ∀ x ∈ spectrum R a, f x ≤ 1) : cfc f a ≤ 1

Proof state:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : OrderedCommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : ∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 1
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } ≤ 1

Tactic:
simpa [ContinuousMap.le_def] using h

Result:
no goals

Proof state:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : OrderedCommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : ∀ (α : Type ?u.715755) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 1
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } ≤ (cfcHom ha) 1

Tactic:
apply cfcHom_mono ha

Result:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : OrderedCommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : ∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 1
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } ≤ 1

Proof state:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : OrderedCommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : ∀ (α : Type ?u.715755) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 1
⊢ 0 ≤ 1

Tactic:
simp

Result:
no goals

Proof state:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : OrderedCommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : ∀ (α : Type ?u.715755) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 1
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } ≤ 1

Tactic:
rw [← map_one (cfcHom ha (R := R))]

Result:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : OrderedCommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : ∀ (α : Type ?u.715755) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 1
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } ≤ (cfcHom ha) 1