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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 28379 | . . 3 ⊢ Rel NrmCVec | |
2 | 1st2nd 7738 | . . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) |
4 | nvop.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 4 | nmcvfval 28384 | . . . 4 ⊢ 𝑁 = (2nd ‘𝑈) |
6 | 5 | opeq2i 4807 | . . 3 ⊢ 〈(1st ‘𝑈), 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
7 | eqid 2821 | . . . . 5 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
8 | nvop.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
9 | nvop.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
10 | 7, 8, 9 | nvvop 28386 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
11 | 10 | opeq1d 4809 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 〈(1st ‘𝑈), 𝑁〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
12 | 6, 11 | syl5eqr 2870 | . 2 ⊢ (𝑈 ∈ NrmCVec → 〈(1st ‘𝑈), (2nd ‘𝑈)〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
13 | 3, 12 | eqtrd 2856 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4573 Rel wrel 5560 ‘cfv 6355 1st c1st 7687 2nd c2nd 7688 NrmCVeccnv 28361 +𝑣 cpv 28362 ·𝑠OLD cns 28364 normCVcnmcv 28367 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-oprab 7160 df-1st 7689 df-2nd 7690 df-vc 28336 df-nv 28369 df-va 28372 df-sm 28374 df-nmcv 28377 |
This theorem is referenced by: sspval 28500 isph 28599 hilhhi 28941 |
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