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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 29502 | . . 3 ⊢ Rel CVecOLD | |
2 | nvss 29535 | . . . . 5 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
3 | nvvop.1 | . . . . . . . 8 ⊢ 𝑊 = (1st ‘𝑈) | |
4 | eqid 2736 | . . . . . . . 8 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
5 | 3, 4 | nvop2 29550 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, (normCV‘𝑈)〉) |
6 | 5 | eleq1d 2822 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec)) |
7 | 6 | ibi 266 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec) |
8 | 2, 7 | sselid 3942 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V)) |
9 | opelxp1 5674 | . . . 4 ⊢ (〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
11 | 1st2nd 7971 | . . 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 29545 | . . . 4 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | 3 | fveq2i 6845 | . . . 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 29547 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
19 | 3 | fveq2i 6845 | . . . 4 ⊢ (2nd ‘𝑊) = (2nd ‘(1st ‘𝑈)) |
20 | 18, 19 | eqtr4i 2767 | . . 3 ⊢ 𝑆 = (2nd ‘𝑊) |
21 | 16, 20 | opeq12i 4835 | . 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 1541 ∈ wcel 2106 Vcvv 3445 〈cop 4592 × cxp 5631 Rel wrel 5638 ‘cfv 6496 1st c1st 7919 2nd c2nd 7920 CVecOLDcvc 29500 NrmCVeccnv 29526 +𝑣 cpv 29527 ·𝑠OLD cns 29529 normCVcnmcv 29532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fo 6502 df-fv 6504 df-oprab 7361 df-1st 7921 df-2nd 7922 df-vc 29501 df-nv 29534 df-va 29537 df-sm 29539 df-nmcv 29542 |
This theorem is referenced by: nvi 29556 nvvc 29557 nvop 29618 |
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