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Mirrors > Home > MPE Home > Th. List > phop | Structured version Visualization version GIF version |
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
phop.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
phop.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
phop.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
phop | ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phrel 28598 | . . 3 ⊢ Rel CPreHilOLD | |
2 | 1st2nd 7720 | . . 3 ⊢ ((Rel CPreHilOLD ∧ 𝑈 ∈ CPreHilOLD) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 689 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) |
4 | phop.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 4 | nmcvfval 28390 | . . . 4 ⊢ 𝑁 = (2nd ‘𝑈) |
6 | 5 | opeq2i 4769 | . . 3 ⊢ 〈(1st ‘𝑈), 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
7 | phnv 28597 | . . . . 5 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
8 | eqid 2798 | . . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
9 | 8 | nvvc 28398 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
10 | vcrel 28343 | . . . . . . 7 ⊢ Rel CVecOLD | |
11 | 1st2nd 7720 | . . . . . . 7 ⊢ ((Rel CVecOLD ∧ (1st ‘𝑈) ∈ CVecOLD) → (1st ‘𝑈) = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉) | |
12 | 10, 11 | mpan 689 | . . . . . 6 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉) |
13 | phop.2 | . . . . . . . 8 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
14 | 13 | vafval 28386 | . . . . . . 7 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | phop.4 | . . . . . . . 8 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
16 | 15 | smfval 28388 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
17 | 14, 16 | opeq12i 4770 | . . . . . 6 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉 |
18 | 12, 17 | eqtr4di 2851 | . . . . 5 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
19 | 7, 9, 18 | 3syl 18 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
20 | 19 | opeq1d 4771 | . . 3 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), 𝑁〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
21 | 6, 20 | syl5eqr 2847 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), (2nd ‘𝑈)〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
22 | 3, 21 | eqtrd 2833 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 〈cop 4531 Rel wrel 5524 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 CVecOLDcvc 28341 NrmCVeccnv 28367 +𝑣 cpv 28368 ·𝑠OLD cns 28370 normCVcnmcv 28373 CPreHilOLDccphlo 28595 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-1st 7671 df-2nd 7672 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-nmcv 28383 df-ph 28596 |
This theorem is referenced by: phpar 28607 |
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