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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 2735 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | nvzcl.6 | . . 3 ⊢ 𝑍 = (0vec‘𝑈) | |
3 | 1, 2 | 0vfval 30635 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
4 | 1 | nvgrp 30646 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp) |
5 | nvzcl.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | 5, 1 | bafval 30633 | . . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
7 | eqid 2735 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
8 | 6, 7 | grpoidcl 30543 | . . 3 ⊢ (( +𝑣 ‘𝑈) ∈ GrpOp → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
9 | 4, 8 | syl 17 | . 2 ⊢ (𝑈 ∈ NrmCVec → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
10 | 3, 9 | eqeltrd 2839 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 GrpOpcgr 30518 GIdcgi 30519 NrmCVeccnv 30613 +𝑣 cpv 30614 BaseSetcba 30615 0veccn0v 30617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-1st 8013 df-2nd 8014 df-grpo 30522 df-gid 30523 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-nmcv 30629 |
This theorem is referenced by: nvmeq0 30687 nvz0 30697 elimnv 30712 nvnd 30717 imsmetlem 30719 dip0r 30746 dip0l 30747 sspz 30764 lno0 30785 lnomul 30789 nvo00 30790 nmosetn0 30794 nmooge0 30796 0oo 30818 0lno 30819 nmoo0 30820 blocni 30834 ubthlem1 30899 minvecolem1 30903 hl0cl 30931 hhshsslem2 31297 |
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