![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nvop | Structured version Visualization version GIF version |
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvop.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvop.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvop.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nvop | ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvrel 30137 | . . 3 ⊢ Rel NrmCVec | |
2 | 1st2nd 8029 | . . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 687 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) |
4 | nvop.6 | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 4 | nmcvfval 30142 | . . . 4 ⊢ 𝑁 = (2nd ‘𝑈) |
6 | 5 | opeq2i 4877 | . . 3 ⊢ 〈(1st ‘𝑈), 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
7 | eqid 2731 | . . . . 5 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
8 | nvop.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
9 | nvop.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
10 | 7, 8, 9 | nvvop 30144 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) = 〈𝐺, 𝑆〉) |
11 | 10 | opeq1d 4879 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 〈(1st ‘𝑈), 𝑁〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
12 | 6, 11 | eqtr3id 2785 | . 2 ⊢ (𝑈 ∈ NrmCVec → 〈(1st ‘𝑈), (2nd ‘𝑈)〉 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
13 | 3, 12 | eqtrd 2771 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 〈cop 4634 Rel wrel 5681 ‘cfv 6543 1st c1st 7977 2nd c2nd 7978 NrmCVeccnv 30119 +𝑣 cpv 30120 ·𝑠OLD cns 30122 normCVcnmcv 30125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-oprab 7416 df-1st 7979 df-2nd 7980 df-vc 30094 df-nv 30127 df-va 30130 df-sm 30132 df-nmcv 30135 |
This theorem is referenced by: sspval 30258 isph 30357 hilhhi 30699 |
Copyright terms: Public domain | W3C validator |