Theorem nvi 31273

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 2763	. . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈)
2		nvi.6	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
3	1, 2	nvop2 31267	. . . . 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 31268	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) = ⟨𝐺, 𝑆⟩)
7	6	opeq1d 4851	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨(1st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
8	3, 7	eqtrd 2798	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
9		id 23	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ NrmCVec)
10	8, 9	eqeltrrd 2865	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec)
11		nvi.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
12	11, 4	bafval 31263	. . . 4 ⊢ 𝑋 = ran 𝐺
13		eqid 2763	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
14	12, 13	isnv 31271	. . 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 31265	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘𝐺))
18	17	eqeq2d 2774	. . . . . 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 1469	. . . 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 3189	. . 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 1471	. 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 2147  ∀wral 3080  ⟨cop 4840   class class class wbr 5108  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  ℂcc 11237  ℝcr 11238  0cc0 11239   + caddc 11242   · cmul 11244   ≤ cle 11383  abscabs 15444  GIdcgi 31149  CVec_OLDcvc 31217  NrmCVeccnv 31243   +_𝑣 cpv 31244  BaseSetcba 31245   ·_𝑠OLD cns 31246  0_veccn0v 31247  norm_CVcnmcv 31249

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-1st 8035  df-2nd 8036  df-vc 31218  df-nv 31251  df-va 31254  df-ba 31255  df-sm 31256  df-0v 31257  df-nmcv 31259

This theorem is used by:  nvvc  31274  nvf  31319  nvs  31322  nvz  31328  nvtri  31329