nvvop - Metamath Proof Explorer

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Theorem nvvop 30589

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 30540	. . 3 ⊢ Rel CVec_OLD
2		nvss 30573	. . . . 5 ⊢ NrmCVec ⊆ (CVec_OLD × V)
3		nvvop.1	. . . . . . . 8 ⊢ 𝑊 = (1^st ‘𝑈)
4		eqid 2731	. . . . . . . 8 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
5	3, 4	nvop2 30588	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (norm_CV‘𝑈)⟩)
6	5	eleq1d 2816	. . . . . 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 3927	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V))
9		opelxp1 5656	. . . 4 ⊢ (⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V) → 𝑊 ∈ CVec_OLD)
10	8, 9	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVec_OLD)
11		1st2nd 7971	. . 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 30583	. . . 4 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15	3	fveq2i 6825	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘(1^st ‘𝑈))
16	14, 15	eqtr4i 2757	. . 3 ⊢ 𝐺 = (1^st ‘𝑊)
17		nvvop.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
18	17	smfval 30585	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
19	3	fveq2i 6825	. . . 4 ⊢ (2^nd ‘𝑊) = (2^nd ‘(1^st ‘𝑈))
20	18, 19	eqtr4i 2757	. . 3 ⊢ 𝑆 = (2^nd ‘𝑊)
21	16, 20	opeq12i 4827	. 2 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩
22	12, 21	eqtr4di 2784	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ⟨cop 4579 × cxp 5612 Rel wrel 5619 ‘cfv 6481 1^st c1st 7919 2^nd c2nd 7920 CVec_OLDcvc 30538 NrmCVeccnv 30564 +_𝑣 cpv 30565 ·_𝑠OLD cns 30567 norm_CVcnmcv 30570

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fo 6487 df-fv 6489 df-oprab 7350 df-1st 7921 df-2nd 7922 df-vc 30539 df-nv 30572 df-va 30575 df-sm 30577 df-nmcv 30580

This theorem is referenced by: nvi 30594 nvvc 30595 nvop 30656

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