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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 28586 | . . 3 ⊢ Rel CPreHilOLD | |
2 | 1st2nd 7732 | . . 3 ⊢ ((Rel CPreHilOLD ∧ 𝑈 ∈ CPreHilOLD) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) |
4 | phop.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 4 | nmcvfval 28378 | . . . 4 ⊢ 𝑁 = (2nd ‘𝑈) |
6 | 5 | opeq2i 4800 | . . 3 ⊢ 〈(1st ‘𝑈), 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
7 | phnv 28585 | . . . . 5 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
8 | eqid 2821 | . . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
9 | 8 | nvvc 28386 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
10 | vcrel 28331 | . . . . . . 7 ⊢ Rel CVecOLD | |
11 | 1st2nd 7732 | . . . . . . 7 ⊢ ((Rel CVecOLD ∧ (1st ‘𝑈) ∈ CVecOLD) → (1st ‘𝑈) = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉) | |
12 | 10, 11 | mpan 688 | . . . . . 6 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉) |
13 | phop.2 | . . . . . . . 8 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
14 | 13 | vafval 28374 | . . . . . . 7 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | phop.4 | . . . . . . . 8 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
16 | 15 | smfval 28376 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
17 | 14, 16 | opeq12i 4801 | . . . . . 6 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉 |
18 | 12, 17 | syl6eqr 2874 | . . . . 5 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
19 | 7, 9, 18 | 3syl 18 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
20 | 19 | opeq1d 4802 | . . 3 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), 𝑁〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
21 | 6, 20 | syl5eqr 2870 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), (2nd ‘𝑈)〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
22 | 3, 21 | eqtrd 2856 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4566 Rel wrel 5554 ‘cfv 6349 1st c1st 7681 2nd c2nd 7682 CVecOLDcvc 28329 NrmCVeccnv 28355 +𝑣 cpv 28356 ·𝑠OLD cns 28358 normCVcnmcv 28361 CPreHilOLDccphlo 28583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-1st 7683 df-2nd 7684 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 df-ph 28584 |
This theorem is referenced by: phpar 28595 |
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