Theorem nvi 29862

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 2732	. . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈)
2		nvi.6	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
3	1, 2	nvop2 29856	. . . . 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 29857	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) = ⟨𝐺, 𝑆⟩)
7	6	opeq1d 4879	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨(1st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
8	3, 7	eqtrd 2772	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
9		id 22	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ NrmCVec)
10	8, 9	eqeltrrd 2834	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec)
11		nvi.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
12	11, 4	bafval 29852	. . . 4 ⊢ 𝑋 = ran 𝐺
13		eqid 2732	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
14	12, 13	isnv 29860	. . 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 29854	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
18	17	eqeq2d 2743	. . . . . 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 1440	. . . 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 3177	. . 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 1442	. 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 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ⟨cop 4634   class class class wbr 5148  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408  1^st c1st 7972  ℂcc 11107  ℝcr 11108  0cc0 11109   + caddc 11112   · cmul 11114   ≤ cle 11248  abscabs 15180  GIdcgi 29738  CVec_OLDcvc 29806  NrmCVeccnv 29832   +_𝑣 cpv 29833  BaseSetcba 29834   ·_𝑠OLD cns 29835  0_veccn0v 29836  norm_CVcnmcv 29838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-1st 7974  df-2nd 7975  df-vc 29807  df-nv 29840  df-va 29843  df-ba 29844  df-sm 29845  df-0v 29846  df-nmcv 29848

This theorem is referenced by:  nvvc  29863  nvf  29908  nvs  29911  nvz  29917  nvtri  29918