-- Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Isometric.lean
lemma nnnorm_cfcₙ_nnreal_lt {f : ℝ≥0 → ℝ≥0} {a : A} {c : ℝ≥0} (h : ∀ x ∈ σₙ ℝ≥0 a, f x < c) :
    ‖cfcₙ f a‖₊ < c

Proof state:
case refine_2
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
⊢ (fun x => ‖x‖₊ < c) ((cfcₙHom ha) { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯, map_zero' := hf0 })

Tactic:
simp only [← cfcₙ_apply f a, (IsGreatest.nnnorm_cfcₙ_nnreal f a hf hf0 ha |>.lt_iff)]

Result:
case refine_2
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
⊢ ∀ x ∈ f '' σₙ ℝ≥0 a, x < c

Proof state:
case refine_2
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
⊢ ∀ x ∈ f '' σₙ ℝ≥0 a, x < c

Tactic:
rintro - ⟨x, hx, rfl⟩

Result:
case refine_2.intro.intro
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
⊢ f x < c

Proof state:
case refine_2.intro.intro
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
⊢ f x < c

Tactic:
exact h x hx

Result:
no goals

Proof state:
case refine_1
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
⊢ (fun x => ‖x‖₊ < c) 0

Tactic:
simpa using zero_le (f 0) |>.trans_lt <| h 0 (quasispectrum.zero_mem ℝ≥0 _)

Result:
no goals