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Mirrors > Home > MPE Home > Th. List > nvzcl | Structured version Visualization version GIF version |
Description: Closure law for the zero vector of a normed complex vector space. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvzcl.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvzcl.6 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
nvzcl | ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | nvzcl.6 | . . 3 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | 0vfval 28389 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
4 | 1 | nvgrp 28400 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp) |
5 | nvzcl.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | 5, 1 | bafval 28387 | . . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
7 | eqid 2798 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
8 | 6, 7 | grpoidcl 28297 | . . 3 ⊢ (( +𝑣 ‘𝑈) ∈ GrpOp → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
9 | 4, 8 | syl 17 | . 2 ⊢ (𝑈 ∈ NrmCVec → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
10 | 3, 9 | eqeltrd 2890 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 GrpOpcgr 28272 GIdcgi 28273 NrmCVeccnv 28367 +𝑣 cpv 28368 BaseSetcba 28369 0veccn0v 28371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-1st 7671 df-2nd 7672 df-grpo 28276 df-gid 28277 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-nmcv 28383 |
This theorem is referenced by: nvmeq0 28441 nvz0 28451 elimnv 28466 nvnd 28471 imsmetlem 28473 dip0r 28500 dip0l 28501 sspz 28518 lno0 28539 lnomul 28543 nvo00 28544 nmosetn0 28548 nmooge0 28550 0oo 28572 0lno 28573 nmoo0 28574 blocni 28588 ubthlem1 28653 minvecolem1 28657 hl0cl 28685 hhshsslem2 29051 |
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