Theorem nvvop 30127

Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvvop.1	⊢ 𝑊 = (1st ‘𝑈)
nvvop.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
nvvop.4	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)

Assertion

Ref	Expression
nvvop	⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Proof of Theorem nvvop

Step	Hyp	Ref	Expression
1		vcrel 30078	. . 3 ⊢ Rel CVecOLD
2		nvss 30111	. . . . 5 ⊢ NrmCVec ⊆ (CVecOLD × V)
3		nvvop.1	. . . . . . . 8 ⊢ 𝑊 = (1st ‘𝑈)
4		eqid 2730	. . . . . . . 8 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
5	3, 4	nvop2 30126	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (normCV‘𝑈)⟩)
6	5	eleq1d 2816	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (normCV‘𝑈)⟩ ∈ NrmCVec))
7	6	ibi 266	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV‘𝑈)⟩ ∈ NrmCVec)
8	2, 7	sselid 3981	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV‘𝑈)⟩ ∈ (CVecOLD × V))
9		opelxp1 5719	. . . 4 ⊢ (⟨𝑊, (normCV‘𝑈)⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
10	8, 9	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
11		1st2nd 8029	. . 3 ⊢ ((Rel CVecOLD ∧ 𝑊 ∈ CVecOLD) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
12	1, 10, 11	sylancr 585	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
13		nvvop.2	. . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
14	13	vafval 30121	. . . 4 ⊢ 𝐺 = (1st ‘(1st ‘𝑈))
15	3	fveq2i 6895	. . . 4 ⊢ (1st ‘𝑊) = (1st ‘(1st ‘𝑈))
16	14, 15	eqtr4i 2761	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
17		nvvop.4	. . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
18	17	smfval 30123	. . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈))
19	3	fveq2i 6895	. . . 4 ⊢ (2nd ‘𝑊) = (2nd ‘(1st ‘𝑈))
20	18, 19	eqtr4i 2761	. . 3 ⊢ 𝑆 = (2nd ‘𝑊)
21	16, 20	opeq12i 4879	. 2 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩
22	12, 21	eqtr4di 2788	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  Vcvv 3472  ⟨cop 4635   × cxp 5675  Rel wrel 5682  ‘cfv 6544  1^st c1st 7977  2^nd c2nd 7978  CVec_OLDcvc 30076  NrmCVeccnv 30102   +_𝑣 cpv 30103   ·_𝑠OLD cns 30105  norm_CVcnmcv 30108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-oprab 7417  df-1st 7979  df-2nd 7980  df-vc 30077  df-nv 30110  df-va 30113  df-sm 30115  df-nmcv 30118

This theorem is referenced by:  nvi  30132  nvvc  30133  nvop  30194