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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 27054 | . . 3 ⊢ Rel CPreHilOLD | |
2 | 1st2nd 7105 | . . 3 ⊢ ((Rel CPreHilOLD ∧ 𝑈 ∈ CPreHilOLD) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 702 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) |
4 | phop.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 4 | nmcvfval 26846 | . . . 4 ⊢ 𝑁 = (2nd ‘𝑈) |
6 | 5 | opeq2i 4344 | . . 3 ⊢ 〈(1st ‘𝑈), 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
7 | phnv 27053 | . . . . 5 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
8 | eqid 2610 | . . . . . 6 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
9 | 8 | nvvc 26854 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
10 | vcrel 26799 | . . . . . . 7 ⊢ Rel CVecOLD | |
11 | 1st2nd 7105 | . . . . . . 7 ⊢ ((Rel CVecOLD ∧ (1st ‘𝑈) ∈ CVecOLD) → (1st ‘𝑈) = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉) | |
12 | 10, 11 | mpan 702 | . . . . . 6 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉) |
13 | phop.2 | . . . . . . . 8 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
14 | 13 | vafval 26842 | . . . . . . 7 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | phop.4 | . . . . . . . 8 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
16 | 15 | smfval 26844 | . . . . . . 7 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
17 | 14, 16 | opeq12i 4345 | . . . . . 6 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))〉 |
18 | 12, 17 | syl6eqr 2662 | . . . . 5 ⊢ ((1st ‘𝑈) ∈ CVecOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
19 | 7, 9, 18 | 3syl 18 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
20 | 19 | opeq1d 4346 | . . 3 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), 𝑁〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
21 | 6, 20 | syl5eqr 2658 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → 〈(1st ‘𝑈), (2nd ‘𝑈)〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
22 | 3, 21 | eqtrd 2644 | 1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 Rel wrel 5043 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 CVecOLDcvc 26797 NrmCVeccnv 26823 +𝑣 cpv 26824 ·𝑠OLD cns 26826 normCVcnmcv 26829 CPreHilOLDccphlo 27051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-1st 7059 df-2nd 7060 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-nmcv 26839 df-ph 27052 |
This theorem is referenced by: phpar 27063 |
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