Theorem isnvi 30632

Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isnvi.5	⊢ 𝑋 = ran 𝐺
isnvi.6	⊢ 𝑍 = (GId‘𝐺)
isnvi.7	⊢ ⟨𝐺, 𝑆⟩ ∈ CVecOLD
isnvi.8	⊢ 𝑁:𝑋⟶ℝ
isnvi.9	⊢ ((𝑥 ∈ 𝑋 ∧ (𝑁‘𝑥) = 0) → 𝑥 = 𝑍)
isnvi.10	⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
isnvi.11	⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
isnvi.12	⊢ 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩

Assertion

Ref	Expression
isnvi	⊢ 𝑈 ∈ NrmCVec

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem isnvi

Step	Hyp	Ref	Expression
1		isnvi.12	. 2 ⊢ 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩
2		isnvi.7	. . 3 ⊢ ⟨𝐺, 𝑆⟩ ∈ CVecOLD
3		isnvi.8	. . 3 ⊢ 𝑁:𝑋⟶ℝ
4		isnvi.9	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑁‘𝑥) = 0) → 𝑥 = 𝑍)
5	4	ex 412	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ((𝑁‘𝑥) = 0 → 𝑥 = 𝑍))
6		isnvi.10	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
7	6	ancoms 458	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
8	7	ralrimiva 3146	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
9		isnvi.11	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
10	9	ralrimiva 3146	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
11	5, 8, 10	3jca 1129	. . . 4 ⊢ (𝑥 ∈ 𝑋 → (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
12	11	rgen 3063	. . 3 ⊢ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
13		isnvi.5	. . . 4 ⊢ 𝑋 = ran 𝐺
14		isnvi.6	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
15	13, 14	isnv 30631	. . 3 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
16	2, 3, 12, 15	mpbir3an 1342	. 2 ⊢ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec
17	1, 16	eqeltri 2837	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⟨cop 4632   class class class wbr 5143  ran crn 5686  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  ℝcr 11154  0cc0 11155   + caddc 11158   · cmul 11160   ≤ cle 11296  abscabs 15273  GIdcgi 30509  CVec_OLDcvc 30577  NrmCVeccnv 30603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-vc 30578  df-nv 30611

This theorem is referenced by:  cnnv  30696  hhnv  31184  hhssnv  31283