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Theorem phop 30838

Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
phop.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
phop.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
phop.6	⊢ 𝑁 = (norm_CV‘𝑈)

**Assertion**
Ref	Expression
phop	⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

**Proof of Theorem phop**
Step	Hyp	Ref	Expression
1		phrel 30835	. . 3 ⊢ Rel CPreHil_OLD
2		1st2nd 8065	. . 3 ⊢ ((Rel CPreHil_OLD ∧ 𝑈 ∈ CPreHil_OLD) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 690	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		phop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 30627	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4876	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		phnv 30834	. . . . 5 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
8		eqid 2736	. . . . . 6 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
9	8	nvvc 30635	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
10		vcrel 30580	. . . . . . 7 ⊢ Rel CVec_OLD
11		1st2nd 8065	. . . . . . 7 ⊢ ((Rel CVec_OLD ∧ (1^st ‘𝑈) ∈ CVec_OLD) → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
12	10, 11	mpan 690	. . . . . 6 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
13		phop.2	. . . . . . . 8 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
14	13	vafval 30623	. . . . . . 7 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15		phop.4	. . . . . . . 8 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
16	15	smfval 30625	. . . . . . 7 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
17	14, 16	opeq12i 4877	. . . . . 6 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩
18	12, 17	eqtr4di 2794	. . . . 5 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
19	7, 9, 18	3syl 18	. . . 4 ⊢ (𝑈 ∈ CPreHil_OLD → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
20	19	opeq1d 4878	. . 3 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
21	6, 20	eqtr3id 2790	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
22	3, 21	eqtrd 2776	1 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⟨cop 4631 Rel wrel 5689 ‘cfv 6560 1^st c1st 8013 2^nd c2nd 8014 CVec_OLDcvc 30578 NrmCVeccnv 30604 +_𝑣 cpv 30605 ·_𝑠OLD cns 30607 norm_CVcnmcv 30610 CPreHil_OLDccphlo 30832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-1st 8015 df-2nd 8016 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-nmcv 30620 df-ph 30833

This theorem is referenced by: phpar 30844

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