Theorem: Differentiable.apply_eq_apply_of_bounded
File: Mathlib/Analysis/Complex/Liouville.lean

tactic: set g : ℂ → F := f ∘ fun t : ℂ => t • (w - z) + z
state_before: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
⊢ f z = f w
state_after: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ f z = f w

tactic: suffices g 0 = g 1 by simpa [g]
state_before: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ f z = f w
state_after: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ g 0 = g 1

tactic: apply liouville_theorem_aux
state_before: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ g 0 = g 1
state_after: case hf
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ Differentiable ℂ g

case hb
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ Bornology.IsBounded (range g)

tactic: exacts [hf.comp ((differentiable_id.smul_const (w - z)).add_const z),
  hb.subset (range_comp_subset_range _ _)]
state_before: case hf
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ Differentiable ℂ g

case hb
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
⊢ Bornology.IsBounded (range g)
state_after: no goals

tactic: simpa [g]
state_before: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
z w : E
g : ℂ → F := f ∘ fun t => t • (w - z) + z
this : g 0 = g 1
⊢ f z = f w
state_after: no goals