Theorem: FormalMultilinearSeries.min_radius_le_radius_add
File: Mathlib/Analysis/Analytic/Basic.lean

tactic: refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type ?u.262305
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p✝ : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
p q : FormalMultilinearSeries 𝕜 E F
⊢ min (radius p) (radius q) ≤ radius (p + q)

tactic: rw [lt_min_iff] at hr
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type ?u.262305
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p✝ : FormalMultilinearSeries 𝕜 E F
r✝ : ℝ≥0
p q : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
hr : ↑r < min (radius p) (radius q)
⊢ ↑r ≤ radius (p + q)

tactic: have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type ?u.262305
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p✝ : FormalMultilinearSeries 𝕜 E F
r✝ : ℝ≥0
p q : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
hr : ↑r < radius p ∧ ↑r < radius q
⊢ ↑r ≤ radius (p + q)

tactic: refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type ?u.262305
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p✝ : FormalMultilinearSeries 𝕜 E F
r✝ : ℝ≥0
p q : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
hr : ↑r < radius p ∧ ↑r < radius q
this : (fun x => ‖p x‖ * ↑r ^ x + ‖q x‖ * ↑r ^ x) =O[atTop] fun x => 1
⊢ ↑r ≤ radius (p + q)

tactic: rw [← add_mul, norm_mul, norm_mul, norm_norm]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type ?u.262305
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p✝ : FormalMultilinearSeries 𝕜 E F
r✝ : ℝ≥0
p q : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
hr : ↑r < radius p ∧ ↑r < radius q
this : (fun x => ‖p x‖ * ↑r ^ x + ‖q x‖ * ↑r ^ x) =O[atTop] fun x => 1
n : ℕ
⊢ ‖‖(p + q) n‖ * ↑r ^ n‖ ≤ ‖‖p n‖ * ↑r ^ n + ‖q n‖ * ↑r ^ n‖

tactic: exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type ?u.262305
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p✝ : FormalMultilinearSeries 𝕜 E F
r✝ : ℝ≥0
p q : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
hr : ↑r < radius p ∧ ↑r < radius q
this : (fun x => ‖p x‖ * ↑r ^ x + ‖q x‖ * ↑r ^ x) =O[atTop] fun x => 1
n : ℕ
⊢ ‖(p + q) n‖ * ‖↑r ^ n‖ ≤ ‖‖p n‖ + ‖q n‖‖ * ‖↑r ^ n‖