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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 2737 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 2 | nvzcl.6 | . . 3 ⊢ 𝑍 = (0vec‘𝑈) | |
| 3 | 1, 2 | 0vfval 30625 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑍 = (GId‘( +𝑣 ‘𝑈))) |
| 4 | 1 | nvgrp 30636 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ GrpOp) |
| 5 | nvzcl.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 6 | 5, 1 | bafval 30623 | . . . 4 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
| 7 | eqid 2737 | . . . 4 ⊢ (GId‘( +𝑣 ‘𝑈)) = (GId‘( +𝑣 ‘𝑈)) | |
| 8 | 6, 7 | grpoidcl 30533 | . . 3 ⊢ (( +𝑣 ‘𝑈) ∈ GrpOp → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
| 9 | 4, 8 | syl 17 | . 2 ⊢ (𝑈 ∈ NrmCVec → (GId‘( +𝑣 ‘𝑈)) ∈ 𝑋) |
| 10 | 3, 9 | eqeltrd 2841 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 GrpOpcgr 30508 GIdcgi 30509 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 0veccn0v 30607 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-1st 8014 df-2nd 8015 df-grpo 30512 df-gid 30513 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 |
| This theorem is referenced by: nvmeq0 30677 nvz0 30687 elimnv 30702 nvnd 30707 imsmetlem 30709 dip0r 30736 dip0l 30737 sspz 30754 lno0 30775 lnomul 30779 nvo00 30780 nmosetn0 30784 nmooge0 30786 0oo 30808 0lno 30809 nmoo0 30810 blocni 30824 ubthlem1 30889 minvecolem1 30893 hl0cl 30921 hhshsslem2 31287 |
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