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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 2825 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | nvzcl.6 | . . 3 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | 0vfval 28016 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
4 | 1 | nvgrp 28027 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp) |
5 | nvzcl.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | 5, 1 | bafval 28014 | . . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
7 | eqid 2825 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
8 | 6, 7 | grpoidcl 27924 | . . 3 ⊢ (( +𝑣 ‘𝑈) ∈ GrpOp → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
9 | 4, 8 | syl 17 | . 2 ⊢ (𝑈 ∈ NrmCVec → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
10 | 3, 9 | eqeltrd 2906 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 GrpOpcgr 27899 GIdcgi 27900 NrmCVeccnv 27994 +𝑣 cpv 27995 BaseSetcba 27996 0veccn0v 27998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-1st 7428 df-2nd 7429 df-grpo 27903 df-gid 27904 df-ablo 27955 df-vc 27969 df-nv 28002 df-va 28005 df-ba 28006 df-sm 28007 df-0v 28008 df-nmcv 28010 |
This theorem is referenced by: nvmeq0 28068 nvz0 28078 elimnv 28093 nvnd 28098 imsmetlem 28100 dip0r 28127 dip0l 28128 sspz 28145 lno0 28166 lnomul 28170 nvo00 28171 nmosetn0 28175 nmooge0 28177 0oo 28199 0lno 28200 nmoo0 28201 blocni 28215 ubthlem1 28281 minvecolem1 28285 hl0cl 28313 hhshsslem2 28680 |
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