![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nvvop | Structured version Visualization version GIF version |
Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvvop.1 | ⊢ 𝑊 = (1st ‘𝑈) |
nvvop.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvvop.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
nvvop | ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈𝐺, 𝑆〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vcrel 30488 | . . 3 ⊢ Rel CVecOLD | |
2 | nvss 30521 | . . . . 5 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
3 | nvvop.1 | . . . . . . . 8 ⊢ 𝑊 = (1st ‘𝑈) | |
4 | eqid 2726 | . . . . . . . 8 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
5 | 3, 4 | nvop2 30536 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, (normCV‘𝑈)〉) |
6 | 5 | eleq1d 2811 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec)) |
7 | 6 | ibi 266 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec) |
8 | 2, 7 | sselid 3977 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V)) |
9 | opelxp1 5715 | . . . 4 ⊢ (〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
11 | 1st2nd 8043 | . . 3 ⊢ ((Rel CVecOLD ∧ 𝑊 ∈ CVecOLD) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
12 | 1, 10, 11 | sylancr 585 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) |
13 | nvvop.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
14 | 13 | vafval 30531 | . . . 4 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | 3 | fveq2i 6894 | . . . 4 ⊢ (1st ‘𝑊) = (1st ‘(1st ‘𝑈)) |
16 | 14, 15 | eqtr4i 2757 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) |
17 | nvvop.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
18 | 17 | smfval 30533 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
19 | 3 | fveq2i 6894 | . . . 4 ⊢ (2nd ‘𝑊) = (2nd ‘(1st ‘𝑈)) |
20 | 18, 19 | eqtr4i 2757 | . . 3 ⊢ 𝑆 = (2nd ‘𝑊) |
21 | 16, 20 | opeq12i 4877 | . 2 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 |
22 | 12, 21 | eqtr4di 2784 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈𝐺, 𝑆〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3463 〈cop 4630 × cxp 5671 Rel wrel 5678 ‘cfv 6544 1st c1st 7991 2nd c2nd 7992 CVecOLDcvc 30486 NrmCVeccnv 30512 +𝑣 cpv 30513 ·𝑠OLD cns 30515 normCVcnmcv 30518 |
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-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7736 |
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-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-oprab 7418 df-1st 7993 df-2nd 7994 df-vc 30487 df-nv 30520 df-va 30523 df-sm 30525 df-nmcv 30528 |
This theorem is referenced by: nvi 30542 nvvc 30543 nvop 30604 |
Copyright terms: Public domain | W3C validator |