Theorem nvi 31013

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 2765	. . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈)
2		nvi.6	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
3	1, 2	nvop2 31007	. . . . 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 31008	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) = ⟨𝐺, 𝑆⟩)
7	6	opeq1d 4846	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨(1st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
8	3, 7	eqtrd 2800	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
9		id 23	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ NrmCVec)
10	8, 9	eqeltrrd 2866	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec)
11		nvi.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
12	11, 4	bafval 31003	. . . 4 ⊢ 𝑋 = ran 𝐺
13		eqid 2765	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
14	12, 13	isnv 31011	. . 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 221	. 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 31005	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
18	17	eqeq2d 2776	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑥 = 𝑍 ↔ 𝑥 = (GId‘𝐺)))
19	18	imbi2d 343	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ↔ ((𝑁‘𝑥) = 0 → 𝑥 = (GId‘𝐺))))
20	19	3anbi1d 1468	. . . 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 3190	. . 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 1470	. 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 260	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)))))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ⟨cop 4597   class class class wbr 5111  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  1^st c1st 7986  ℂcc 11109  ℝcr 11110  0cc0 11111   + caddc 11114   · cmul 11116   ≤ cle 11255  abscabs 15305  GIdcgi 30889  CVec_OLDcvc 30957  NrmCVeccnv 30983   +_𝑣 cpv 30984  BaseSetcba 30985   ·_𝑠OLD cns 30986  0_veccn0v 30987  norm_CVcnmcv 30989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-1st 7988  df-2nd 7989  df-vc 30958  df-nv 30991  df-va 30994  df-ba 30995  df-sm 30996  df-0v 30997  df-nmcv 30999

This theorem is used by:  nvvc  31014  nvf  31059  nvs  31062  nvz  31068  nvtri  31069