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Mirrors > Home > MPE Home > Th. List > nvop | Structured version Visualization version GIF version |
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvop.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvop.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvop.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nvop | ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvrel 26841 | . . 3 ⊢ Rel NrmCVec | |
2 | 1st2nd 7105 | . . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 702 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) |
4 | nvop.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 4 | nmcvfval 26846 | . . . 4 ⊢ 𝑁 = (2nd ‘𝑈) |
6 | 5 | opeq2i 4344 | . . 3 ⊢ 〈(1st ‘𝑈), 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
7 | eqid 2610 | . . . . 5 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
8 | nvop.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
9 | nvop.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
10 | 7, 8, 9 | nvvop 26848 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
11 | 10 | opeq1d 4346 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 〈(1st ‘𝑈), 𝑁〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
12 | 6, 11 | syl5eqr 2658 | . 2 ⊢ (𝑈 ∈ NrmCVec → 〈(1st ‘𝑈), (2nd ‘𝑈)〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
13 | 3, 12 | eqtrd 2644 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 Rel wrel 5043 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 NrmCVeccnv 26823 +𝑣 cpv 26824 ·𝑠OLD cns 26826 normCVcnmcv 26829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-oprab 6553 df-1st 7059 df-2nd 7060 df-vc 26798 df-nv 26831 df-va 26834 df-sm 26836 df-nmcv 26839 |
This theorem is referenced by: sspval 26962 isph 27061 hilhhi 27405 |
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