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Theorem nvvop 26848

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 = (1^st ‘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 26799	. . 3 ⊢ Rel CVec_OLD
2		nvss 26832	. . . . 5 ⊢ NrmCVec ⊆ (CVec_OLD × V)
3		nvvop.1	. . . . . . . 8 ⊢ 𝑊 = (1^st ‘𝑈)
4		eqid 2610	. . . . . . . 8 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
5	3, 4	nvop2 26847	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (norm_CV‘𝑈)⟩)
6	5	eleq1d 2672	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ NrmCVec))
7	6	ibi 255	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ NrmCVec)
8	2, 7	sseldi 3566	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V))
9		opelxp1 5074	. . . 4 ⊢ (⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V) → 𝑊 ∈ CVec_OLD)
10	8, 9	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVec_OLD)
11		1st2nd 7105	. . 3 ⊢ ((Rel CVec_OLD ∧ 𝑊 ∈ CVec_OLD) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
12	1, 10, 11	sylancr 694	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
13		nvvop.2	. . . . 5 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
14	13	vafval 26842	. . . 4 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15	3	fveq2i 6106	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘(1^st ‘𝑈))
16	14, 15	eqtr4i 2635	. . 3 ⊢ 𝐺 = (1^st ‘𝑊)
17		nvvop.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
18	17	smfval 26844	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
19	3	fveq2i 6106	. . . 4 ⊢ (2^nd ‘𝑊) = (2^nd ‘(1^st ‘𝑈))
20	18, 19	eqtr4i 2635	. . 3 ⊢ 𝑆 = (2^nd ‘𝑊)
21	16, 20	opeq12i 4345	. 2 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩
22	12, 21	syl6eqr 2662	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⟨cop 4131 × cxp 5036 Rel wrel 5043 ‘cfv 5804 1^st c1st 7057 2^nd c2nd 7058 CVec_OLDcvc 26797 NrmCVeccnv 26823 +_𝑣 cpv 26824 ·_𝑠OLD cns 26826 norm_CVcnmcv 26829

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-oprab 6553 df-1st 7059 df-2nd 7060 df-vc 26798 df-nv 26831 df-va 26834 df-sm 26836 df-nmcv 26839

This theorem is referenced by: nvi 26853 nvvc 26854 nvop 26915

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