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Mirrors > Home > MPE Home > Th. List > grpinvex | Structured version Visualization version Unicode version |
Description: Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grpcl.b | |
grpcl.p | |
grpinvex.p |
Ref | Expression |
---|---|
grpinvex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpcl.b | . . . 4 | |
2 | grpcl.p | . . . 4 | |
3 | grpinvex.p | . . . 4 | |
4 | 1, 2, 3 | isgrp 17428 | . . 3 |
5 | 4 | simprbi 480 | . 2 |
6 | oveq2 6658 | . . . . 5 | |
7 | 6 | eqeq1d 2624 | . . . 4 |
8 | 7 | rexbidv 3052 | . . 3 |
9 | 8 | rspccva 3308 | . 2 |
10 | 5, 9 | sylan 488 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 wrex 2913 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cmnd 17294 cgrp 17422 |
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-ral 2917 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-iota 5851 df-fv 5896 df-ov 6653 df-grp 17425 |
This theorem is referenced by: dfgrp2 17447 grprcan 17455 grpinveu 17456 grprinv 17469 |
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