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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 30805 | . . 3 ⊢ Rel NrmCVec | |
| 2 | 1st2nd 8020 | . . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉) | |
| 3 | 1, 2 | mpan 700 | . 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 30810 | . . 3 ⊢ 𝑁 = (2nd ‘𝑈) |
| 7 | 4, 6 | opeq12i 4836 | . 2 ⊢ 〈𝑊, 𝑁〉 = 〈(1st ‘𝑈), (2nd ‘𝑈)〉 |
| 8 | 3, 7 | eqtr4di 2815 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, 𝑁〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1560 ∈ wcel 2142 〈cop 4588 Rel wrel 5652 ‘cfv 6521 1st c1st 7968 2nd c2nd 7969 NrmCVeccnv 30787 normCVcnmcv 30793 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pr 5390 ax-un 7718 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-iota 6477 df-fun 6523 df-fv 6529 df-oprab 7400 df-1st 7970 df-2nd 7971 df-nv 30795 df-nmcv 30803 |
| This theorem is referenced by: nvvop 30812 nvi 30817 |
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