Theorem: Entire.nontrivialAnalyticOn (Ray/AnalyticManifold/Nontrivial.lean)

State before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ
s : Set ℂ
fa : AnalyticOn ℂ f univ
ne : ∃ a b, f a ≠ f b
⊢ NontrivialAnalyticOn f univ
Tactic:
refine isPreconnected_univ.nontrivialAnalyticOn fa ?_
State after:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ
s : Set ℂ
fa : AnalyticOn ℂ f univ
ne : ∃ a b, f a ≠ f b
⊢ ∃ a b, a ∈ univ ∧ b ∈ univ ∧ f a ≠ f b

State before:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → ℂ
s : Set ℂ
fa : AnalyticOn ℂ f univ
ne : ∃ a b, f a ≠ f b
⊢ ∃ a b, a ∈ univ ∧ b ∈ univ ∧ f a ≠ f b
Tactic:
simpa only [Set.mem_univ, true_and_iff]
State after:
no goals