Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > grpinvval2 | Structured version Visualization version Unicode version |
Description: A df-neg 10269-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
grpsubcl.b | |
grpsubcl.m | |
grpinvsub.n | |
grpinvval2.z |
Ref | Expression |
---|---|
grpinvval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubcl.b | . . . 4 | |
2 | grpinvval2.z | . . . 4 | |
3 | 1, 2 | grpidcl 17450 | . . 3 |
4 | eqid 2622 | . . . 4 | |
5 | grpinvsub.n | . . . 4 | |
6 | grpsubcl.m | . . . 4 | |
7 | 1, 4, 5, 6 | grpsubval 17465 | . . 3 |
8 | 3, 7 | sylan 488 | . 2 |
9 | 1, 5 | grpinvcl 17467 | . . 3 |
10 | 1, 4, 2 | grplid 17452 | . . 3 |
11 | 9, 10 | syldan 487 | . 2 |
12 | 8, 11 | eqtr2d 2657 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cgrp 17422 cminusg 17423 csg 17424 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 |
This theorem is referenced by: grpsubadd0sub 17502 matinvgcell 20241 istgp2 21895 nrmmetd 22379 nminv 22425 |
Copyright terms: Public domain | W3C validator |