Theorem: IsBoundedBilinearMap.continuous
File: Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean

tactic: refine continuous_iff_continuousAt.2 fun x ↦ tendsto_sub_nhds_zero_iff.1 ?_
state_before: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
⊢ Continuous f

tactic: suffices : Tendsto (λ y : E × F ↦ f (y.1 - x.1, y.2) + f (x.1, y.2 - x.2)) (𝓝 x) (𝓝 (0 + 0))
state_before: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Tendsto (fun n => f n - f x) (𝓝 x) (𝓝 0)

tactic: apply Tendsto.add
state_before: case this
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Tendsto (fun y => f (y.fst - x.fst, y.snd) + f (x.fst, y.snd - x.snd)) (𝓝 x) (𝓝 (0 + 0))

tactic: simpa only [h.map_sub_left, h.map_sub_right, sub_add_sub_cancel, zero_add] using this
state_before: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
this : Tendsto (fun y => f (y.fst - x.fst, y.snd) + f (x.fst, y.snd - x.snd)) (𝓝 x) (𝓝 (0 + 0))
⊢ Tendsto (fun n => f n - f x) (𝓝 x) (𝓝 0)

tactic: rw [← isLittleO_one_iff ℝ, ← one_mul 1]
state_before: case this.hf
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Tendsto (fun x_1 => f (x_1.fst - x.fst, x_1.snd)) (𝓝 x) (𝓝 0)

tactic: refine h.isBigO_comp.trans_isLittleO ?_
state_before: case this.hf
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ (fun x_1 => f (x_1.fst - x.fst, x_1.snd)) =o[𝓝 x] fun _x => 1 * 1

tactic: refine (IsLittleO.norm_left ?_).mul_isBigO (IsBigO.norm_left ?_)
state_before: case this.hf
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ (fun x_1 => ‖x_1.fst - x.fst‖ * ‖x_1.snd‖) =o[𝓝 x] fun _x => 1 * 1

tactic: exact (isLittleO_one_iff _).2 (tendsto_sub_nhds_zero_iff.2 (continuous_fst.tendsto _))
state_before: case this.hf.refine_1
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ (fun x_1 => x_1.fst - x.fst) =o[𝓝 x] fun _x => 1

tactic: exact (continuous_snd.tendsto _).isBigO_one ℝ
state_before: case this.hf.refine_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ (fun x => x.snd) =O[𝓝 x] fun _x => 1

tactic: refine Continuous.tendsto' ?_ _ _ (by rw [h.map_sub_right, sub_self])
state_before: case this.hg
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Tendsto (fun x_1 => f (x.fst, x_1.snd - x.snd)) (𝓝 x) (𝓝 0)

tactic: exact ((h.toContinuousLinearMap x.1).continuous).comp (continuous_snd.sub continuous_const)
state_before: case this.hg
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Continuous fun x_1 => f (x.fst, x_1.snd - x.snd)

tactic: rw [h.map_sub_right, sub_self]
state_before: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ f (x.fst, x.snd - x.snd) = 0