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Theorem nvop 31278

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 31204	. . 3 ⊢ Rel NrmCVec
2		1st2nd 8050	. . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 703	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		nvop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 31209	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4837	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		eqid 2761	. . . . 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 31211	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
11	10	opeq1d 4839	. . 3 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
12	6, 11	eqtr3id 2810	. 2 ⊢ (𝑈 ∈ NrmCVec → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
13	3, 12	eqtrd 2796	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 Rel wrel 5656 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000 NrmCVeccnv 31186 +_𝑣 cpv 31187 ·_𝑠OLD cns 31189 norm_CVcnmcv 31192

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fo 6544 df-fv 6546 df-oprab 7424 df-1st 8001 df-2nd 8002 df-vc 31161 df-nv 31194 df-va 31197 df-sm 31199 df-nmcv 31202

This theorem is used by: sspval 31325 isph 31424 hilhhi 31766

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