Theorem isnvi 30641

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 3143	. . . . 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 3143	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
11	5, 8, 10	3jca 1127	. . . 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 3060	. . 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 30640	. . 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 1340	. 2 ⊢ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec
17	1, 16	eqeltri 2834	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ⟨cop 4636   class class class wbr 5147  ran crn 5689  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  ℂcc 11150  ℝcr 11151  0cc0 11152   + caddc 11155   · cmul 11157   ≤ cle 11293  abscabs 15269  GIdcgi 30518  CVec_OLDcvc 30586  NrmCVeccnv 30612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-vc 30587  df-nv 30620

This theorem is referenced by:  cnnv  30705  hhnv  31193  hhssnv  31292