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Mirrors > Home > MPE Home > Th. List > ngpgrp | Structured version Visualization version Unicode version |
Description: A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpgrp | NrmGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 | |
2 | eqid 2622 | . . 3 | |
3 | eqid 2622 | . . 3 | |
4 | 1, 2, 3 | isngp 22400 | . 2 NrmGrp |
5 | 4 | simp1bi 1076 | 1 NrmGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1990 wss 3574 ccom 5118 cfv 5888 cds 15950 cgrp 17422 csg 17424 cmt 22123 cnm 22381 NrmGrpcngp 22382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-co 5123 df-iota 5851 df-fv 5896 df-ngp 22388 |
This theorem is referenced by: ngpds 22408 ngpds2 22410 ngpds3 22412 ngprcan 22414 isngp4 22416 ngpinvds 22417 ngpsubcan 22418 nmf 22419 nmge0 22421 nmeq0 22422 nminv 22425 nmmtri 22426 nmsub 22427 nmrtri 22428 nm2dif 22429 nmtri 22430 nmtri2 22431 ngpi 22432 nm0 22433 ngptgp 22440 tngngp2 22456 tnggrpr 22459 nrmtngnrm 22462 nlmdsdi 22485 nlmdsdir 22486 nrginvrcnlem 22495 ngpocelbl 22508 nmo0 22539 nmotri 22543 0nghm 22545 nmoid 22546 idnghm 22547 nmods 22548 nmcn 22647 nmoleub2lem2 22916 nmhmcn 22920 cphipval2 23040 4cphipval2 23041 cphipval 23042 ipcnlem2 23043 nglmle 23100 qqhcn 30035 |
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