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Theorem phop 31170

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 31167	. . 3 ⊢ Rel CPreHil_OLD
2		1st2nd 8032	. . 3 ⊢ ((Rel CPreHil_OLD ∧ 𝑈 ∈ CPreHil_OLD) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 702	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		phop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 30959	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4842	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		phnv 31166	. . . . 5 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
8		eqid 2763	. . . . . 6 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
9	8	nvvc 30967	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
10		vcrel 30912	. . . . . . 7 ⊢ Rel CVec_OLD
11		1st2nd 8032	. . . . . . 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 702	. . . . . 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 30955	. . . . . . 7 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15		phop.4	. . . . . . . 8 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
16	15	smfval 30957	. . . . . . 7 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
17	14, 16	opeq12i 4843	. . . . . 6 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩
18	12, 17	eqtr4di 2816	. . . . 5 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
19	7, 9, 18	3syl 19	. . . 4 ⊢ (𝑈 ∈ CPreHil_OLD → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
20	19	opeq1d 4844	. . 3 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
21	6, 20	eqtr3id 2812	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
22	3, 21	eqtrd 2798	1 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1570 ∈ wcel 2143 ⟨cop 4595 Rel wrel 5666 ‘cfv 6536 1^st c1st 7980 2^nd c2nd 7981 CVec_OLDcvc 30910 NrmCVeccnv 30936 +_𝑣 cpv 30937 ·_𝑠OLD cns 30939 norm_CVcnmcv 30942 CPreHil_OLDccphlo 31164

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-1st 7982 df-2nd 7983 df-vc 30911 df-nv 30944 df-va 30947 df-ba 30948 df-sm 30949 df-0v 30950 df-nmcv 30952 df-ph 31165

This theorem is referenced by: phpar 31176

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