Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ ∅, 0 < g' x

pp: case refine_2
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ ⦃s t : Set X⦄,
    s ⊆ t →
      (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x) →
        ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x

pp: case refine_3
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ ⦃s t : Set X⦄,
    (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x) →
      (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x) →
        ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s ∪ t, 0 < g' x

pp: case refine_4
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ x ∈ t, ∃ t_1 ∈ nhdsWithin x t, ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t_1, 0 < g' x

Tactic:
· refine ⟨0, ?_, fun x hx => False.elim hx⟩
      convert! I.zero_mem
      ext
      simp only [comp_apply, zero_apply, ContinuousMap.coe_coe, map_zero]

Goal state after:
case refine_2
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ ⦃s t : Set X⦄,
    s ⊆ t →
      (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x) →
        ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x

case refine_3
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ ⦃s t : Set X⦄,
    (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x) →
      (∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x) →
        ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s ∪ t, 0 < g' x

case refine_4
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ x ∈ t, ∃ t_1 ∈ nhdsWithin x t, ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t_1, 0 < g' x