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Theorem nvop 30766

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 30692	. . 3 ⊢ Rel NrmCVec
2		1st2nd 7987	. . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 691	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		nvop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 30697	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4821	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		eqid 2737	. . . . 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 30699	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
11	10	opeq1d 4823	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
12	6, 11	eqtr3id 2786	. 2 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
13	3, 12	eqtrd 2772	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4574 Rel wrel 5631 ‘cfv 6494 1^st c1st 7935 2^nd c2nd 7936 NrmCVeccnv 30674 +_𝑣 cpv 30675 ·_𝑠OLD cns 30677 norm_CVcnmcv 30680

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-sep 5232 ax-nul 5242 ax-pr 5372 ax-un 7684

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-rab 3391 df-v 3432 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-br 5087 df-opab 5149 df-mpt 5168 df-id 5521 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-fo 6500 df-fv 6502 df-oprab 7366 df-1st 7937 df-2nd 7938 df-vc 30649 df-nv 30682 df-va 30685 df-sm 30687 df-nmcv 30690

This theorem is referenced by: sspval 30813 isph 30912 hilhhi 31254

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