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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 30532 | . . 3 ⊢ Rel NrmCVec | |
2 | 1st2nd 8045 | . . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
3 | 1, 2 | mpan 688 | . 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 30537 | . . 3 ⊢ 𝑁 = (2nd ‘𝑈) |
7 | 4, 6 | opeq12i 4876 | . 2 ⊢ 〈𝑊, 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
8 | 3, 7 | eqtr4di 2784 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 〈cop 4629 Rel wrel 5679 ‘cfv 6546 1st c1st 7993 2nd c2nd 7994 NrmCVeccnv 30514 normCVcnmcv 30520 |
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 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
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-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-iota 6498 df-fun 6548 df-fv 6554 df-oprab 7420 df-1st 7995 df-2nd 7996 df-nv 30522 df-nmcv 30530 |
This theorem is referenced by: nvvop 30539 nvi 30544 |
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