Theorem: JoinedIn.of_segment_subset
File: Mathlib/Analysis/Convex/Topology.lean

Proof steps:
state_before: ι : Type u_1
𝕜 : Type u_2
E✝ : Type u_3
inst✝⁹ : AddCommGroup E✝
inst✝⁸ : Module ℝ E✝
inst✝⁷ : TopologicalSpace E✝
inst✝⁶ : TopologicalAddGroup E✝
inst✝⁵ : ContinuousSMul ℝ E✝
E : Type u_4
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
x y : E
s : Set E
h : [x-[ℝ]y] ⊆ s
⊢ JoinedIn s x y
state_after: ι : Type u_1
𝕜 : Type u_2
E✝ : Type u_3
inst✝⁹ : AddCommGroup E✝
inst✝⁸ : Module ℝ E✝
inst✝⁷ : TopologicalSpace E✝
inst✝⁶ : TopologicalAddGroup E✝
inst✝⁵ : ContinuousSMul ℝ E✝
E : Type u_4
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
x y : E
s : Set E
h : [x-[ℝ]y] ⊆ s
A : Continuous fun t => (1 - t) • x + t • y
⊢ JoinedIn s x y
tactic: have A : Continuous (fun t ↦ (1 - t) • x + t • y : ℝ → E) := by fun_prop
state_before: ι : Type u_1
𝕜 : Type u_2
E✝ : Type u_3
inst✝⁹ : AddCommGroup E✝
inst✝⁸ : Module ℝ E✝
inst✝⁷ : TopologicalSpace E✝
inst✝⁶ : TopologicalAddGroup E✝
inst✝⁵ : ContinuousSMul ℝ E✝
E : Type u_4
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
x y : E
s : Set E
h : [x-[ℝ]y] ⊆ s
A : Continuous fun t => (1 - t) • x + t • y
⊢ JoinedIn s x y
state_after: ι : Type u_1
𝕜 : Type u_2
E✝ : Type u_3
inst✝⁹ : AddCommGroup E✝
inst✝⁸ : Module ℝ E✝
inst✝⁷ : TopologicalSpace E✝
inst✝⁶ : TopologicalAddGroup E✝
inst✝⁵ : ContinuousSMul ℝ E✝
E : Type u_4
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
x y : E
s : Set E
h : [x-[ℝ]y] ⊆ s
A : Continuous fun t => (1 - t) • x + t • y
⊢ (fun t => (1 - t) • x + t • y) '' unitInterval ⊆ s
tactic: apply JoinedIn.ofLine A.continuousOn (by simp) (by simp)
state_before: ι : Type u_1
𝕜 : Type u_2
E✝ : Type u_3
inst✝⁹ : AddCommGroup E✝
inst✝⁸ : Module ℝ E✝
inst✝⁷ : TopologicalSpace E✝
inst✝⁶ : TopologicalAddGroup E✝
inst✝⁵ : ContinuousSMul ℝ E✝
E : Type u_4
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
x y : E
s : Set E
h : [x-[ℝ]y] ⊆ s
A : Continuous fun t => (1 - t) • x + t • y
⊢ (fun t => (1 - t) • x + t • y) '' unitInterval ⊆ s
state_after: case h.e'_3
ι : Type u_1
𝕜 : Type u_2
E✝ : Type u_3
inst✝⁹ : AddCommGroup E✝
inst✝⁸ : Module ℝ E✝
inst✝⁷ : TopologicalSpace E✝
inst✝⁶ : TopologicalAddGroup E✝
inst✝⁵ : ContinuousSMul ℝ E✝
E : Type u_4
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
x y : E
s : Set E
h : [x-[ℝ]y] ⊆ s
A : Continuous fun t => (1 - t) • x + t • y
⊢ (fun t => (1 - t) • x + t • y) '' unitInterval = [x-[ℝ]y]
tactic: convert h
state_before: case h.e'_3
ι : Type u_1
𝕜 : Type u_2
E✝ : Type u_3
inst✝⁹ : AddCommGroup E✝
inst✝⁸ : Module ℝ E✝
inst✝⁷ : TopologicalSpace E✝
inst✝⁶ : TopologicalAddGroup E✝
inst✝⁵ : ContinuousSMul ℝ E✝
E : Type u_4
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
x y : E
s : Set E
h : [x-[ℝ]y] ⊆ s
A : Continuous fun t => (1 - t) • x + t • y
⊢ (fun t => (1 - t) • x + t • y) '' unitInterval = [x-[ℝ]y]
state_after: no goals
tactic: rw [segment_eq_image ℝ x y]