Theorem nvi 28955

Description: The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvi.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvi.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
nvi.4	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
nvi.5	⊢ 𝑍 = (0vec‘𝑈)
nvi.6	⊢ 𝑁 = (normCV‘𝑈)

Assertion

Ref	Expression
nvi	⊢ (𝑈 ∈ 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)))))

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

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

Proof of Theorem nvi

Step	Hyp	Ref	Expression
1		eqid 2739	. . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈)
2		nvi.6	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
3	1, 2	nvop2 28949	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st ‘𝑈), 𝑁⟩)
4		nvi.2	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
5		nvi.4	. . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
6	1, 4, 5	nvvop 28950	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) = ⟨𝐺, 𝑆⟩)
7	6	opeq1d 4815	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨(1st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
8	3, 7	eqtrd 2779	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
9		id 22	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ NrmCVec)
10	8, 9	eqeltrrd 2841	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec)
11		nvi.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
12	11, 4	bafval 28945	. . . 4 ⊢ 𝑋 = ran 𝐺
13		eqid 2739	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
14	12, 13	isnv 28953	. . 3 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
15	10, 14	sylib 217	. 2 ⊢ (𝑈 ∈ NrmCVec → (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
16		nvi.5	. . . . . . . 8 ⊢ 𝑍 = (0vec‘𝑈)
17	4, 16	0vfval 28947	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
18	17	eqeq2d 2750	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑥 = 𝑍 ↔ 𝑥 = (GId‘𝐺)))
19	18	imbi2d 340	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ↔ ((𝑁‘𝑥) = 0 → 𝑥 = (GId‘𝐺))))
20	19	3anbi1d 1438	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ((((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ↔ (((𝑁‘𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
21	20	ralbidv 3122	. . 3 ⊢ (𝑈 ∈ NrmCVec → (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ↔ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
22	21	3anbi3d 1440	. 2 ⊢ (𝑈 ∈ 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)))) ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (GId‘𝐺)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
23	15, 22	mpbird 256	1 ⊢ (𝑈 ∈ 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)))))

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065  ⟨cop 4572   class class class wbr 5078  ⟶wf 6426  ‘cfv 6430  (class class class)co 7268  1^st c1st 7815  ℂcc 10853  ℝcr 10854  0cc0 10855   + caddc 10858   · cmul 10860   ≤ cle 10994  abscabs 14926  GIdcgi 28831  CVec_OLDcvc 28899  NrmCVeccnv 28925   +_𝑣 cpv 28926  BaseSetcba 28927   ·_𝑠OLD cns 28928  0_veccn0v 28929  norm_CVcnmcv 28931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-1st 7817  df-2nd 7818  df-vc 28900  df-nv 28933  df-va 28936  df-ba 28937  df-sm 28938  df-0v 28939  df-nmcv 28941

This theorem is referenced by:  nvvc  28956  nvf  29001  nvs  29004  nvz  29010  nvtri  29011