Theorem isnvi 30645

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 3152	. . . . 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 3152	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
11	5, 8, 10	3jca 1128	. . . 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 3069	. . 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 30644	. . 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 1341	. 2 ⊢ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec
17	1, 16	eqeltri 2840	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ⟨cop 4654   class class class wbr 5166  ran crn 5701  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  ℝcr 11183  0cc0 11184   + caddc 11187   · cmul 11189   ≤ cle 11325  abscabs 15283  GIdcgi 30522  CVec_OLDcvc 30590  NrmCVeccnv 30616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-vc 30591  df-nv 30624

This theorem is referenced by:  cnnv  30709  hhnv  31197  hhssnv  31296