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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 2738 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | nvzcl.6 | . . 3 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | 0vfval 28533 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
4 | 1 | nvgrp 28544 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp) |
5 | nvzcl.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | 5, 1 | bafval 28531 | . . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
7 | eqid 2738 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
8 | 6, 7 | grpoidcl 28441 | . . 3 ⊢ (( +𝑣 ‘𝑈) ∈ GrpOp → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
9 | 4, 8 | syl 17 | . 2 ⊢ (𝑈 ∈ NrmCVec → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
10 | 3, 9 | eqeltrd 2833 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 GrpOpcgr 28416 GIdcgi 28417 NrmCVeccnv 28511 +𝑣 cpv 28512 BaseSetcba 28513 0veccn0v 28515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 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 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-1st 7707 df-2nd 7708 df-grpo 28420 df-gid 28421 df-ablo 28472 df-vc 28486 df-nv 28519 df-va 28522 df-ba 28523 df-sm 28524 df-0v 28525 df-nmcv 28527 |
This theorem is referenced by: nvmeq0 28585 nvz0 28595 elimnv 28610 nvnd 28615 imsmetlem 28617 dip0r 28644 dip0l 28645 sspz 28662 lno0 28683 lnomul 28687 nvo00 28688 nmosetn0 28692 nmooge0 28694 0oo 28716 0lno 28717 nmoo0 28718 blocni 28732 ubthlem1 28797 minvecolem1 28801 hl0cl 28829 hhshsslem2 29195 |
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