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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 28373 | . . 3 ⊢ Rel NrmCVec | |
2 | 1st2nd 7732 | . . 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 28378 | . . 3 ⊢ 𝑁 = (2nd ‘𝑈) |
7 | 4, 6 | opeq12i 4801 | . 2 ⊢ 〈𝑊, 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
8 | 3, 7 | syl6eqr 2874 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, 𝑁〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4566 Rel wrel 5554 ‘cfv 6349 1st c1st 7681 2nd c2nd 7682 NrmCVeccnv 28355 normCVcnmcv 28361 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-oprab 7154 df-1st 7683 df-2nd 7684 df-nv 28363 df-nmcv 28371 |
This theorem is referenced by: nvvop 28380 nvi 28385 |
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