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Theorem nvop 30656

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 30582	. . 3 ⊢ Rel NrmCVec
2		1st2nd 7971	. . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 690	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		nvop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 30587	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4826	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		eqid 2731	. . . . 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 30589	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
11	10	opeq1d 4828	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
12	6, 11	eqtr3id 2780	. 2 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
13	3, 12	eqtrd 2766	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ⟨cop 4579 Rel wrel 5619 ‘cfv 6481 1^st c1st 7919 2^nd c2nd 7920 NrmCVeccnv 30564 +_𝑣 cpv 30565 ·_𝑠OLD cns 30567 norm_CVcnmcv 30570

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fo 6487 df-fv 6489 df-oprab 7350 df-1st 7921 df-2nd 7922 df-vc 30539 df-nv 30572 df-va 30575 df-sm 30577 df-nmcv 30580

This theorem is referenced by: sspval 30703 isph 30802 hilhhi 31144

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