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Theorem phop 30847

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 30844	. . 3 ⊢ Rel CPreHil_OLD
2		1st2nd 8063	. . 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 30636	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4882	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		phnv 30843	. . . . 5 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
8		eqid 2735	. . . . . 6 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
9	8	nvvc 30644	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
10		vcrel 30589	. . . . . . 7 ⊢ Rel CVec_OLD
11		1st2nd 8063	. . . . . . 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 30632	. . . . . . 7 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15		phop.4	. . . . . . . 8 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
16	15	smfval 30634	. . . . . . 7 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
17	14, 16	opeq12i 4883	. . . . . 6 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩
18	12, 17	eqtr4di 2793	. . . . 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 4884	. . 3 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
21	6, 20	eqtr3id 2789	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
22	3, 21	eqtrd 2775	1 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⟨cop 4637 Rel wrel 5694 ‘cfv 6563 1^st c1st 8011 2^nd c2nd 8012 CVec_OLDcvc 30587 NrmCVeccnv 30613 +_𝑣 cpv 30614 ·_𝑠OLD cns 30616 norm_CVcnmcv 30619 CPreHil_OLDccphlo 30841

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-1st 8013 df-2nd 8014 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-nmcv 30629 df-ph 30842

This theorem is referenced by: phpar 30853

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