nvvop - Metamath Proof Explorer

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Theorem nvvop 30553

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 30504	. . 3 ⊢ Rel CVec_OLD
2		nvss 30537	. . . . 5 ⊢ NrmCVec ⊆ (CVec_OLD × V)
3		nvvop.1	. . . . . . . 8 ⊢ 𝑊 = (1^st ‘𝑈)
4		eqid 2729	. . . . . . . 8 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
5	3, 4	nvop2 30552	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (norm_CV‘𝑈)⟩)
6	5	eleq1d 2813	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ NrmCVec))
7	6	ibi 267	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ NrmCVec)
8	2, 7	sselid 3933	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V))
9		opelxp1 5661	. . . 4 ⊢ (⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V) → 𝑊 ∈ CVec_OLD)
10	8, 9	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVec_OLD)
11		1st2nd 7974	. . 3 ⊢ ((Rel CVec_OLD ∧ 𝑊 ∈ CVec_OLD) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
12	1, 10, 11	sylancr 587	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
13		nvvop.2	. . . . 5 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
14	13	vafval 30547	. . . 4 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15	3	fveq2i 6825	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘(1^st ‘𝑈))
16	14, 15	eqtr4i 2755	. . 3 ⊢ 𝐺 = (1^st ‘𝑊)
17		nvvop.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
18	17	smfval 30549	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
19	3	fveq2i 6825	. . . 4 ⊢ (2^nd ‘𝑊) = (2^nd ‘(1^st ‘𝑈))
20	18, 19	eqtr4i 2755	. . 3 ⊢ 𝑆 = (2^nd ‘𝑊)
21	16, 20	opeq12i 4829	. 2 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩
22	12, 21	eqtr4di 2782	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3436 ⟨cop 4583 × cxp 5617 Rel wrel 5624 ‘cfv 6482 1^st c1st 7922 2^nd c2nd 7923 CVec_OLDcvc 30502 NrmCVeccnv 30528 +_𝑣 cpv 30529 ·_𝑠OLD cns 30531 norm_CVcnmcv 30534

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-fo 6488 df-fv 6490 df-oprab 7353 df-1st 7924 df-2nd 7925 df-vc 30503 df-nv 30536 df-va 30539 df-sm 30541 df-nmcv 30544

This theorem is referenced by: nvi 30558 nvvc 30559 nvop 30620

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