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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 30748 | . . 3 ⊢ Rel CPreHilOLD | |
2 | 1st2nd 8053 | . . 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 30540 | . . . 4 ⊢ 𝑁 = (2nd ‘𝑈) |
6 | 5 | opeq2i 4883 | . . 3 ⊢ 〈(1st ‘𝑈), 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
7 | phnv 30747 | . . . . 5 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
8 | eqid 2726 | . . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
9 | 8 | nvvc 30548 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
10 | vcrel 30493 | . . . . . . 7 ⊢ Rel CVecOLD | |
11 | 1st2nd 8053 | . . . . . . 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 30536 | . . . . . . 7 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | phop.4 | . . . . . . . 8 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
16 | 15 | smfval 30538 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
17 | 14, 16 | opeq12i 4884 | . . . . . 6 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉 |
18 | 12, 17 | eqtr4di 2784 | . . . . 5 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
19 | 7, 9, 18 | 3syl 18 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
20 | 19 | opeq1d 4885 | . . 3 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), 𝑁〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
21 | 6, 20 | eqtr3id 2780 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), (2nd ‘𝑈)〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
22 | 3, 21 | eqtrd 2766 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 〈cop 4639 Rel wrel 5687 ‘cfv 6554 1st c1st 8001 2nd c2nd 8002 CVecOLDcvc 30491 NrmCVeccnv 30517 +𝑣 cpv 30518 ·𝑠OLD cns 30520 normCVcnmcv 30523 CPreHilOLDccphlo 30745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-1st 8003 df-2nd 8004 df-vc 30492 df-nv 30525 df-va 30528 df-ba 30529 df-sm 30530 df-0v 30531 df-nmcv 30533 df-ph 30746 |
This theorem is referenced by: phpar 30757 |
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