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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 2725 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
2 | eqid 2725 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2725 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | 1, 2, 3 | isngp 24549 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
5 | 4 | simp1bi 1142 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3944 ∘ ccom 5682 ‘cfv 6549 distcds 17245 Grpcgrp 18898 -gcsg 18900 MetSpcms 24268 normcnm 24529 NrmGrpcngp 24530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-co 5687 df-iota 6501 df-fv 6557 df-ngp 24536 |
This theorem is referenced by: ngpds 24557 ngpds2 24559 ngpds3 24561 ngprcan 24563 isngp4 24565 ngpinvds 24566 ngpsubcan 24567 nmf 24568 nmge0 24570 nmeq0 24571 nminv 24574 nmmtri 24575 nmsub 24576 nmrtri 24577 nm2dif 24578 nmtri 24579 nmtri2 24580 ngpi 24581 nm0 24582 ngptgp 24589 tngngp2 24613 tnggrpr 24616 nrmtngnrm 24619 nlmdsdi 24642 nlmdsdir 24643 nrginvrcnlem 24652 ngpocelbl 24665 nmo0 24696 nmotri 24700 0nghm 24702 nmoid 24703 idnghm 24704 nmods 24705 nmcn 24804 nmoleub2lem2 25087 nmhmcn 25091 cphpyth 25188 cphipval2 25213 4cphipval2 25214 cphipval 25215 ipcnlem2 25216 nglmle 25274 qqhcn 33723 |
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