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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 28641 | . . 3 ⊢ Rel CVecOLD | |
2 | nvss 28674 | . . . . 5 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
3 | nvvop.1 | . . . . . . . 8 ⊢ 𝑊 = (1st ‘𝑈) | |
4 | eqid 2737 | . . . . . . . 8 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
5 | 3, 4 | nvop2 28689 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, (normCV‘𝑈)〉) |
6 | 5 | eleq1d 2822 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec)) |
7 | 6 | ibi 270 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec) |
8 | 2, 7 | sseldi 3899 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V)) |
9 | opelxp1 5592 | . . . 4 ⊢ (〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
11 | 1st2nd 7810 | . . 3 ⊢ ((Rel CVecOLD ∧ 𝑊 ∈ CVecOLD) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
12 | 1, 10, 11 | sylancr 590 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) |
13 | nvvop.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
14 | 13 | vafval 28684 | . . . 4 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | 3 | fveq2i 6720 | . . . 4 ⊢ (1st ‘𝑊) = (1st ‘(1st ‘𝑈)) |
16 | 14, 15 | eqtr4i 2768 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) |
17 | nvvop.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
18 | 17 | smfval 28686 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
19 | 3 | fveq2i 6720 | . . . 4 ⊢ (2nd ‘𝑊) = (2nd ‘(1st ‘𝑈)) |
20 | 18, 19 | eqtr4i 2768 | . . 3 ⊢ 𝑆 = (2nd ‘𝑊) |
21 | 16, 20 | opeq12i 4789 | . 2 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 |
22 | 12, 21 | eqtr4di 2796 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈𝐺, 𝑆〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Vcvv 3408 〈cop 4547 × cxp 5549 Rel wrel 5556 ‘cfv 6380 1st c1st 7759 2nd c2nd 7760 CVecOLDcvc 28639 NrmCVeccnv 28665 +𝑣 cpv 28666 ·𝑠OLD cns 28668 normCVcnmcv 28671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fo 6386 df-fv 6388 df-oprab 7217 df-1st 7761 df-2nd 7762 df-vc 28640 df-nv 28673 df-va 28676 df-sm 28678 df-nmcv 28681 |
This theorem is referenced by: nvi 28695 nvvc 28696 nvop 28757 |
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