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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 28519 | . . 3 ⊢ Rel CPreHilOLD | |
2 | 1st2nd 7727 | . . 3 ⊢ ((Rel CPreHilOLD ∧ 𝑈 ∈ CPreHilOLD) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 686 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) |
4 | phop.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 4 | nmcvfval 28311 | . . . 4 ⊢ 𝑁 = (2nd ‘𝑈) |
6 | 5 | opeq2i 4799 | . . 3 ⊢ 〈(1st ‘𝑈), 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
7 | phnv 28518 | . . . . 5 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
8 | eqid 2818 | . . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
9 | 8 | nvvc 28319 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
10 | vcrel 28264 | . . . . . . 7 ⊢ Rel CVecOLD | |
11 | 1st2nd 7727 | . . . . . . 7 ⊢ ((Rel CVecOLD ∧ (1st ‘𝑈) ∈ CVecOLD) → (1st ‘𝑈) = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉) | |
12 | 10, 11 | mpan 686 | . . . . . 6 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉) |
13 | phop.2 | . . . . . . . 8 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
14 | 13 | vafval 28307 | . . . . . . 7 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | phop.4 | . . . . . . . 8 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
16 | 15 | smfval 28309 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
17 | 14, 16 | opeq12i 4800 | . . . . . 6 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉 |
18 | 12, 17 | syl6eqr 2871 | . . . . 5 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
19 | 7, 9, 18 | 3syl 18 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
20 | 19 | opeq1d 4801 | . . 3 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), 𝑁〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
21 | 6, 20 | syl5eqr 2867 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), (2nd ‘𝑈)〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
22 | 3, 21 | eqtrd 2853 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 〈cop 4563 Rel wrel 5553 ‘cfv 6348 1st c1st 7676 2nd c2nd 7677 CVecOLDcvc 28262 NrmCVeccnv 28288 +𝑣 cpv 28289 ·𝑠OLD cns 28291 normCVcnmcv 28294 CPreHilOLDccphlo 28516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-1st 7678 df-2nd 7679 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-nmcv 28304 df-ph 28517 |
This theorem is referenced by: phpar 28528 |
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