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Theorem nvop 29038

Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

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

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

**Proof of Theorem nvop**
Step	Hyp	Ref	Expression
1		nvrel 28964	. . 3 ⊢ Rel NrmCVec
2		1st2nd 7880	. . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 687	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		nvop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 28969	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4808	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		eqid 2738	. . . . 5 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
8		nvop.2	. . . . 5 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
9		nvop.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
10	7, 8, 9	nvvop 28971	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
11	10	opeq1d 4810	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
12	6, 11	eqtr3id 2792	. 2 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
13	3, 12	eqtrd 2778	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ⟨cop 4567 Rel wrel 5594 ‘cfv 6433 1^st c1st 7829 2^nd c2nd 7830 NrmCVeccnv 28946 +_𝑣 cpv 28947 ·_𝑠OLD cns 28949 norm_CVcnmcv 28952

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fo 6439 df-fv 6441 df-oprab 7279 df-1st 7831 df-2nd 7832 df-vc 28921 df-nv 28954 df-va 28957 df-sm 28959 df-nmcv 28962

This theorem is referenced by: sspval 29085 isph 29184 hilhhi 29526

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