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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 30580 | . . 3 ⊢ Rel CVecOLD | |
| 2 | nvss 30613 | . . . . 5 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
| 3 | nvvop.1 | . . . . . . . 8 ⊢ 𝑊 = (1st ‘𝑈) | |
| 4 | eqid 2736 | . . . . . . . 8 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
| 5 | 3, 4 | nvop2 30628 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, (normCV‘𝑈)〉) | 
| 6 | 5 | eleq1d 2825 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec)) | 
| 7 | 6 | ibi 267 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec) | 
| 8 | 2, 7 | sselid 3980 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V)) | 
| 9 | opelxp1 5726 | . . . 4 ⊢ (〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD) | |
| 10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) | 
| 11 | 1st2nd 8065 | . . 3 ⊢ ((Rel CVecOLD ∧ 𝑊 ∈ CVecOLD) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
| 12 | 1, 10, 11 | sylancr 587 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | 
| 13 | nvvop.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
| 14 | 13 | vafval 30623 | . . . 4 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) | 
| 15 | 3 | fveq2i 6908 | . . . 4 ⊢ (1st ‘𝑊) = (1st ‘(1st ‘𝑈)) | 
| 16 | 14, 15 | eqtr4i 2767 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) | 
| 17 | nvvop.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
| 18 | 17 | smfval 30625 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) | 
| 19 | 3 | fveq2i 6908 | . . . 4 ⊢ (2nd ‘𝑊) = (2nd ‘(1st ‘𝑈)) | 
| 20 | 18, 19 | eqtr4i 2767 | . . 3 ⊢ 𝑆 = (2nd ‘𝑊) | 
| 21 | 16, 20 | opeq12i 4877 | . 2 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 | 
| 22 | 12, 21 | eqtr4di 2794 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈𝐺, 𝑆〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 〈cop 4631 × cxp 5682 Rel wrel 5689 ‘cfv 6560 1st c1st 8013 2nd c2nd 8014 CVecOLDcvc 30578 NrmCVeccnv 30604 +𝑣 cpv 30605 ·𝑠OLD cns 30607 normCVcnmcv 30610 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-oprab 7436 df-1st 8015 df-2nd 8016 df-vc 30579 df-nv 30612 df-va 30615 df-sm 30617 df-nmcv 30620 | 
| This theorem is referenced by: nvi 30634 nvvc 30635 nvop 30696 | 
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