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Theorem nvop 30969

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 30895	. . 3 ⊢ Rel NrmCVec
2		1st2nd 8036	. . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 702	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		nvop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 30900	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4846	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		eqid 2769	. . . . 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 30902	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
11	10	opeq1d 4848	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
12	6, 11	eqtr3id 2818	. 2 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
13	3, 12	eqtrd 2804	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1567 ∈ wcel 2149 ⟨cop 4600 Rel wrel 5667 ‘cfv 6537 1^st c1st 7984 2^nd c2nd 7985 NrmCVeccnv 30877 +_𝑣 cpv 30878 ·_𝑠OLD cns 30880 norm_CVcnmcv 30883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-oprab 7415 df-1st 7986 df-2nd 7987 df-vc 30852 df-nv 30885 df-va 30888 df-sm 30890 df-nmcv 30893

This theorem is referenced by: sspval 31016 isph 31115 hilhhi 31457

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