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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 30349 | . . 3 ⊢ Rel NrmCVec | |
2 | 1st2nd 8019 | . . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩) | |
3 | 1, 2 | mpan 687 | . 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 30354 | . . 3 ⊢ 𝑁 = (2nd ‘𝑈) |
7 | 4, 6 | opeq12i 4871 | . 2 ⊢ ⟨𝑊, 𝑁⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩ |
8 | 3, 7 | eqtr4di 2782 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⟨cop 4627 Rel wrel 5672 ‘cfv 6534 1st c1st 7967 2nd c2nd 7968 NrmCVeccnv 30331 normCVcnmcv 30337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fv 6542 df-oprab 7406 df-1st 7969 df-2nd 7970 df-nv 30339 df-nmcv 30347 |
This theorem is referenced by: nvvop 30356 nvi 30361 |
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