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Mirrors > Home > MPE Home > Th. List > nvop2 | Structured version Visualization version GIF version |
Description: A normed complex vector space is an ordered pair of a vector space and a norm operation. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvop2.1 | ⊢ 𝑊 = (1st ‘𝑈) |
nvop2.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nvop2 | ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, 𝑁〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvrel 30631 | . . 3 ⊢ Rel NrmCVec | |
2 | 1st2nd 8063 | . . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) |
4 | nvop2.1 | . . 3 ⊢ 𝑊 = (1st ‘𝑈) | |
5 | nvop2.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
6 | 5 | nmcvfval 30636 | . . 3 ⊢ 𝑁 = (2nd ‘𝑈) |
7 | 4, 6 | opeq12i 4883 | . 2 ⊢ 〈𝑊, 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
8 | 3, 7 | eqtr4di 2793 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 〈cop 4637 Rel wrel 5694 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 NrmCVeccnv 30613 normCVcnmcv 30619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-oprab 7435 df-1st 8013 df-2nd 8014 df-nv 30621 df-nmcv 30629 |
This theorem is referenced by: nvvop 30638 nvi 30643 |
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