feat: lemma concerning for ‖cfc f a‖ < c (#18361)

Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Isometric.lean

@@ -106,6 +106,20 @@ lemma norm_cfc_le_iff (f : 𝕜 → 𝕜) (a : A) {c : ℝ} (hc : 0 ≤ c)
     (ha : p a := by cfc_tac) : ‖cfc f a‖ ≤ c ↔ ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c :=
   ⟨fun h _ hx ↦ norm_apply_le_norm_cfc f a hx hf ha |>.trans h, norm_cfc_le hc⟩
 
+lemma norm_cfc_lt {f : 𝕜 → 𝕜} {a : A} {c : ℝ} (hc : 0 < c) (h : ∀ x ∈ σ 𝕜 a, ‖f x‖ < c) :
+    ‖cfc f a‖ < c := by
+  obtain (_ | _) := subsingleton_or_nontrivial A
+  · simpa [Subsingleton.elim (cfc f a) 0]
+  · refine cfc_cases (‖·‖ < c) a f (by simpa) fun hf ha ↦ ?_
+    simp only [← cfc_apply f a, (IsGreatest.norm_cfc f a hf ha |>.lt_iff)]
+    rintro - ⟨x, hx, rfl⟩
+    exact h x hx
+
+lemma norm_cfc_lt_iff (f : 𝕜 → 𝕜) (a : A) {c : ℝ} (hc : 0 < c)
+    (hf : ContinuousOn f (σ 𝕜 a) := by cfc_cont_tac)
+    (ha : p a := by cfc_tac) : ‖cfc f a‖ < c ↔ ∀ x ∈ σ 𝕜 a, ‖f x‖ < c :=
+  ⟨fun h _ hx ↦ norm_apply_le_norm_cfc f a hx hf ha |>.trans_lt h, norm_cfc_lt hc⟩
+
 open NNReal
 
 lemma nnnorm_cfc_le {f : 𝕜 → 𝕜} {a : A} (c : ℝ≥0) (h : ∀ x ∈ σ 𝕜 a, ‖f x‖₊ ≤ c) :
@@ -117,6 +131,15 @@ lemma nnnorm_cfc_le_iff (f : 𝕜 → 𝕜) (a : A) (c : ℝ≥0)
     (ha : p a := by cfc_tac) : ‖cfc f a‖₊ ≤ c ↔ ∀ x ∈ σ 𝕜 a, ‖f x‖₊ ≤ c :=
   norm_cfc_le_iff f a c.2
 
+lemma nnnorm_cfc_lt {f : 𝕜 → 𝕜} {a : A} {c : ℝ≥0} (hc : 0 < c) (h : ∀ x ∈ σ 𝕜 a, ‖f x‖₊ < c) :
+    ‖cfc f a‖₊ < c :=
+  norm_cfc_lt hc h
+
+lemma nnnorm_cfc_lt_iff (f : 𝕜 → 𝕜) (a : A) {c : ℝ≥0} (hc : 0 < c)
+    (hf : ContinuousOn f (σ 𝕜 a) := by cfc_cont_tac)
+    (ha : p a := by cfc_tac) : ‖cfc f a‖₊ < c ↔ ∀ x ∈ σ 𝕜 a, ‖f x‖₊ < c :=
+  norm_cfc_lt_iff f a hc
+
 end NormedRing
 
 namespace SpectrumRestricts
@@ -251,6 +274,19 @@ lemma norm_cfcₙ_le_iff (f : 𝕜 → 𝕜) (a : A) (c : ℝ)
     (ha : p a := by cfc_tac) : ‖cfcₙ f a‖ ≤ c ↔ ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c :=
   ⟨fun h _ hx ↦ norm_apply_le_norm_cfcₙ f a hx hf hf₀ ha |>.trans h, norm_cfcₙ_le⟩
 
+lemma norm_cfcₙ_lt {f : 𝕜 → 𝕜} {a : A} {c : ℝ} (h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c) :
+    ‖cfcₙ f a‖ < c := by
+  refine cfcₙ_cases (‖·‖ < c) a f ?_ fun hf hf0 ha ↦ ?_
+  · simpa using (norm_nonneg _).trans_lt <| h 0 (quasispectrum.zero_mem 𝕜 a)
+  · simp only [← cfcₙ_apply f a, (IsGreatest.norm_cfcₙ f a hf hf0 ha |>.lt_iff)]
+    rintro - ⟨x, hx, rfl⟩
+    exact h x hx
+
+lemma norm_cfcₙ_lt_iff (f : 𝕜 → 𝕜) (a : A) (c : ℝ)
+    (hf : ContinuousOn f (σₙ 𝕜 a) := by cfc_cont_tac) (hf₀ : f 0 = 0 := by cfc_zero_tac)
+    (ha : p a := by cfc_tac) : ‖cfcₙ f a‖ < c ↔ ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c :=
+  ⟨fun h _ hx ↦ norm_apply_le_norm_cfcₙ f a hx hf hf₀ ha |>.trans_lt h, norm_cfcₙ_lt⟩
+
 open NNReal
 
 lemma nnnorm_cfcₙ_le {f : 𝕜 → 𝕜} {a : A} {c : ℝ≥0} (h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖₊ ≤ c) :
@@ -262,6 +298,15 @@ lemma nnnorm_cfcₙ_le_iff (f : 𝕜 → 𝕜) (a : A) (c : ℝ≥0)
     (ha : p a := by cfc_tac) : ‖cfcₙ f a‖₊ ≤ c ↔ ∀ x ∈ σₙ 𝕜 a, ‖f x‖₊ ≤ c :=
   norm_cfcₙ_le_iff f a c.1 hf hf₀ ha
 
+lemma nnnorm_cfcₙ_lt {f : 𝕜 → 𝕜} {a : A} {c : ℝ≥0} (h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖₊ < c) :
+    ‖cfcₙ f a‖₊ < c :=
+  norm_cfcₙ_lt h
+
+lemma nnnorm_cfcₙ_lt_iff (f : 𝕜 → 𝕜) (a : A) (c : ℝ≥0)
+    (hf : ContinuousOn f (σₙ 𝕜 a) := by cfc_cont_tac) (hf₀ : f 0 = 0 := by cfc_zero_tac)
+    (ha : p a := by cfc_tac) : ‖cfcₙ f a‖₊ < c ↔ ∀ x ∈ σₙ 𝕜 a, ‖f x‖₊ < c :=
+  norm_cfcₙ_lt_iff f a c.1 hf hf₀ ha
+
 end NormedRing
 
 namespace QuasispectrumRestricts
@@ -442,6 +487,21 @@ lemma nnnorm_cfc_nnreal_le_iff (f : ℝ≥0 → ℝ≥0) (a : A) (c : ℝ≥0)
     (ha : 0 ≤ a := by cfc_tac) : ‖cfc f a‖₊ ≤ c ↔ ∀ x ∈ σ ℝ≥0 a, f x ≤ c :=
   ⟨fun h _ hx ↦ apply_le_nnnorm_cfc_nnreal f a hx hf ha |>.trans h, nnnorm_cfc_nnreal_le⟩
 
+lemma nnnorm_cfc_nnreal_lt {f : ℝ≥0 → ℝ≥0} {a : A} {c : ℝ≥0} (hc : 0 < c)
+    (h : ∀ x ∈ σ ℝ≥0 a, f x < c) : ‖cfc f a‖₊ < c := by
+  obtain (_ | _) := subsingleton_or_nontrivial A
+  · rw [Subsingleton.elim (cfc f a) 0]
+    simpa
+  · refine cfc_cases (‖·‖₊ < c) a f (by simpa) fun hf ha ↦ ?_
+    simp only [← cfc_apply f a, (IsGreatest.nnnorm_cfc_nnreal f a hf ha |>.lt_iff)]
+    rintro - ⟨x, hx, rfl⟩
+    exact h x hx
+
+lemma nnnorm_cfc_nnreal_lt_iff (f : ℝ≥0 → ℝ≥0) (a : A) {c : ℝ≥0} (hc : 0 < c)
+    (hf : ContinuousOn f (σ ℝ≥0 a) := by cfc_cont_tac)
+    (ha : 0 ≤ a := by cfc_tac) : ‖cfc f a‖₊ < c ↔ ∀ x ∈ σ ℝ≥0 a, f x < c :=
+  ⟨fun h _ hx ↦ apply_le_nnnorm_cfc_nnreal f a hx hf ha |>.trans_lt h, nnnorm_cfc_nnreal_lt hc⟩
+
 end Unital
 
 section NonUnital
@@ -483,6 +543,19 @@ lemma nnnorm_cfcₙ_nnreal_le_iff (f : ℝ≥0 → ℝ≥0) (a : A) (c : ℝ≥0
     (ha : 0 ≤ a := by cfc_tac) : ‖cfcₙ f a‖₊ ≤ c ↔ ∀ x ∈ σₙ ℝ≥0 a, f x ≤ c :=
   ⟨fun h _ hx ↦ apply_le_nnnorm_cfcₙ_nnreal f a hx hf hf₀ ha |>.trans h, nnnorm_cfcₙ_nnreal_le⟩
 
+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 := by
+  refine cfcₙ_cases (‖·‖₊ < c) a f ?_ fun hf hf0 ha ↦ ?_
+  · simpa using zero_le (f 0) |>.trans_lt <| h 0 (quasispectrum.zero_mem ℝ≥0 _)
+  · simp only [← cfcₙ_apply f a, (IsGreatest.nnnorm_cfcₙ_nnreal f a hf hf0 ha |>.lt_iff)]
+    rintro - ⟨x, hx, rfl⟩
+    exact h x hx
+
+lemma nnnorm_cfcₙ_nnreal_lt_iff (f : ℝ≥0 → ℝ≥0) (a : A) (c : ℝ≥0)
+    (hf : ContinuousOn f (σₙ ℝ≥0 a) := by cfc_cont_tac) (hf₀ : f 0 = 0 := by cfc_zero_tac)
+    (ha : 0 ≤ a := by cfc_tac) : ‖cfcₙ f a‖₊ < c ↔ ∀ x ∈ σₙ ℝ≥0 a, f x < c :=
+  ⟨fun h _ hx ↦ apply_le_nnnorm_cfcₙ_nnreal f a hx hf hf₀ ha |>.trans_lt h, nnnorm_cfcₙ_nnreal_lt⟩
+
 end NonUnital
 
 end NNReal