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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 | β’ π = (BaseSetβπ) |
| nvzcl.6 | β’ π = (0vecβπ) |
| Ref | Expression |
|---|---|
| nvzcl | β’ (π β NrmCVec β π β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2733 | . . 3 β’ ( +π£ βπ) = ( +π£ βπ) | |
| 2 | nvzcl.6 | . . 3 β’ π = (0vecβπ) | |
| 3 | 1, 2 | 0vfval 29859 | . 2 β’ (π β NrmCVec β π = (GIdβ( +π£ βπ))) |
| 4 | 1 | nvgrp 29870 | . . 3 β’ (π β NrmCVec β ( +π£ βπ) β GrpOp) |
| 5 | nvzcl.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 6 | 5, 1 | bafval 29857 | . . . 4 β’ π = ran ( +π£ βπ) |
| 7 | eqid 2733 | . . . 4 β’ (GIdβ( +π£ βπ)) = (GIdβ( +π£ βπ)) | |
| 8 | 6, 7 | grpoidcl 29767 | . . 3 β’ (( +π£ βπ) β GrpOp β (GIdβ( +π£ βπ)) β π) |
| 9 | 4, 8 | syl 17 | . 2 β’ (π β NrmCVec β (GIdβ( +π£ βπ)) β π) |
| 10 | 3, 9 | eqeltrd 2834 | 1 β’ (π β NrmCVec β π β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6544 GrpOpcgr 29742 GIdcgi 29743 NrmCVeccnv 29837 +π£ cpv 29838 BaseSetcba 29839 0veccn0v 29841 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-1st 7975 df-2nd 7976 df-grpo 29746 df-gid 29747 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-nmcv 29853 |
| This theorem is referenced by: nvmeq0 29911 nvz0 29921 elimnv 29936 nvnd 29941 imsmetlem 29943 dip0r 29970 dip0l 29971 sspz 29988 lno0 30009 lnomul 30013 nvo00 30014 nmosetn0 30018 nmooge0 30020 0oo 30042 0lno 30043 nmoo0 30044 blocni 30058 ubthlem1 30123 minvecolem1 30127 hl0cl 30155 hhshsslem2 30521 |
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