nvvop - Metamath Proof Explorer

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Theorem nvvop 27773

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 27724	. . 3 ⊢ Rel CVec_OLD
2		nvss 27757	. . . . 5 ⊢ NrmCVec ⊆ (CVec_OLD × V)
3		nvvop.1	. . . . . . . 8 ⊢ 𝑊 = (1^st ‘𝑈)
4		eqid 2760	. . . . . . . 8 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
5	3, 4	nvop2 27772	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (norm_CV‘𝑈)⟩)
6	5	eleq1d 2824	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ NrmCVec))
7	6	ibi 256	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ NrmCVec)
8	2, 7	sseldi 3742	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V))
9		opelxp1 5307	. . . 4 ⊢ (⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V) → 𝑊 ∈ CVec_OLD)
10	8, 9	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVec_OLD)
11		1st2nd 7381	. . 3 ⊢ ((Rel CVec_OLD ∧ 𝑊 ∈ CVec_OLD) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
12	1, 10, 11	sylancr 698	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
13		nvvop.2	. . . . 5 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
14	13	vafval 27767	. . . 4 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15	3	fveq2i 6355	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘(1^st ‘𝑈))
16	14, 15	eqtr4i 2785	. . 3 ⊢ 𝐺 = (1^st ‘𝑊)
17		nvvop.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
18	17	smfval 27769	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
19	3	fveq2i 6355	. . . 4 ⊢ (2^nd ‘𝑊) = (2^nd ‘(1^st ‘𝑈))
20	18, 19	eqtr4i 2785	. . 3 ⊢ 𝑆 = (2^nd ‘𝑊)
21	16, 20	opeq12i 4558	. 2 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩
22	12, 21	syl6eqr 2812	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⟨cop 4327 × cxp 5264 Rel wrel 5271 ‘cfv 6049 1^st c1st 7331 2^nd c2nd 7332 CVec_OLDcvc 27722 NrmCVeccnv 27748 +_𝑣 cpv 27749 ·_𝑠OLD cns 27751 norm_CVcnmcv 27754

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-fo 6055 df-fv 6057 df-oprab 6817 df-1st 7333 df-2nd 7334 df-vc 27723 df-nv 27756 df-va 27759 df-sm 27761 df-nmcv 27764

This theorem is referenced by: nvi 27778 nvvc 27779 nvop 27840

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