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Theorem phop 30907

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 30904	. . 3 ⊢ Rel CPreHil_OLD
2		1st2nd 7986	. . 3 ⊢ ((Rel CPreHil_OLD ∧ 𝑈 ∈ CPreHil_OLD) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 691	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		phop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 30696	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4821	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		phnv 30903	. . . . 5 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
8		eqid 2737	. . . . . 6 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
9	8	nvvc 30704	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
10		vcrel 30649	. . . . . . 7 ⊢ Rel CVec_OLD
11		1st2nd 7986	. . . . . . 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 691	. . . . . 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 30692	. . . . . . 7 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15		phop.4	. . . . . . . 8 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
16	15	smfval 30694	. . . . . . 7 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
17	14, 16	opeq12i 4822	. . . . . 6 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩
18	12, 17	eqtr4di 2790	. . . . 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 4823	. . 3 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
21	6, 20	eqtr3id 2786	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
22	3, 21	eqtrd 2772	1 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4574 Rel wrel 5630 ‘cfv 6493 1^st c1st 7934 2^nd c2nd 7935 CVec_OLDcvc 30647 NrmCVeccnv 30673 +_𝑣 cpv 30674 ·_𝑠OLD cns 30676 norm_CVcnmcv 30679 CPreHil_OLDccphlo 30901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pr 5371 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-oprab 7365 df-1st 7936 df-2nd 7937 df-vc 30648 df-nv 30681 df-va 30684 df-ba 30685 df-sm 30686 df-0v 30687 df-nmcv 30689 df-ph 30902

This theorem is referenced by: phpar 30913

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