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| Mirrors > Home > MPE Home > Th. List > ngpgrp | Structured version Visualization version GIF version | ||
| Description: A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| ngpgrp | ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
| 2 | eqid 2737 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
| 3 | eqid 2737 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
| 4 | 1, 2, 3 | isngp 24609 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
| 5 | 4 | simp1bi 1146 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ⊆ wss 3951 ∘ ccom 5689 ‘cfv 6561 distcds 17306 Grpcgrp 18951 -gcsg 18953 MetSpcms 24328 normcnm 24589 NrmGrpcngp 24590 |
| 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-ext 2708 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 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-br 5144 df-opab 5206 df-co 5694 df-iota 6514 df-fv 6569 df-ngp 24596 |
| This theorem is referenced by: ngpds 24617 ngpds2 24619 ngpds3 24621 ngprcan 24623 isngp4 24625 ngpinvds 24626 ngpsubcan 24627 nmf 24628 nmge0 24630 nmeq0 24631 nminv 24634 nmmtri 24635 nmsub 24636 nmrtri 24637 nm2dif 24638 nmtri 24639 nmtri2 24640 ngpi 24641 nm0 24642 ngptgp 24649 tngngp2 24673 tnggrpr 24676 nrmtngnrm 24679 nlmdsdi 24702 nlmdsdir 24703 nrginvrcnlem 24712 ngpocelbl 24725 nmo0 24756 nmotri 24760 0nghm 24762 nmoid 24763 idnghm 24764 nmods 24765 nmcn 24866 nmoleub2lem2 25149 nmhmcn 25153 cphpyth 25250 cphipval2 25275 4cphipval2 25276 cphipval 25277 ipcnlem2 25278 nglmle 25336 qqhcn 33992 |
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