Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > grpodivf | Structured version Visualization version Unicode version |
Description: Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpdivf.1 | |
grpdivf.3 |
Ref | Expression |
---|---|
grpodivf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpdivf.1 | . . . . . . . 8 | |
2 | eqid 2622 | . . . . . . . 8 | |
3 | 1, 2 | grpoinvcl 27378 | . . . . . . 7 |
4 | 3 | 3adant2 1080 | . . . . . 6 |
5 | 1 | grpocl 27354 | . . . . . 6 |
6 | 4, 5 | syld3an3 1371 | . . . . 5 |
7 | 6 | 3expib 1268 | . . . 4 |
8 | 7 | ralrimivv 2970 | . . 3 |
9 | eqid 2622 | . . . 4 | |
10 | 9 | fmpt2 7237 | . . 3 |
11 | 8, 10 | sylib 208 | . 2 |
12 | grpdivf.3 | . . . 4 | |
13 | 1, 2, 12 | grpodivfval 27388 | . . 3 |
14 | 13 | feq1d 6030 | . 2 |
15 | 11, 14 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 wral 2912 cxp 5112 crn 5115 wf 5884 cfv 5888 (class class class)co 6650 cmpt2 6652 cgr 27343 cgn 27345 cgs 27346 |
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-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-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 |
This theorem is referenced by: grpodivcl 27393 |
Copyright terms: Public domain | W3C validator |