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Mirrors > Home > ILE Home > Th. List > grprinvd | Unicode version |
Description: Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grprinvlem.c | |
grprinvlem.o | |
grprinvlem.i | |
grprinvlem.a | |
grprinvlem.n | |
grprinvd.x | |
grprinvd.n | |
grprinvd.e |
Ref | Expression |
---|---|
grprinvd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grprinvlem.c | . 2 | |
2 | grprinvlem.o | . 2 | |
3 | grprinvlem.i | . 2 | |
4 | grprinvlem.a | . 2 | |
5 | grprinvlem.n | . 2 | |
6 | 1 | 3expb 1105 | . . . . 5 |
7 | 6 | caovclg 5653 | . . . 4 |
8 | 7 | adantlr 446 | . . 3 |
9 | grprinvd.x | . . 3 | |
10 | grprinvd.n | . . 3 | |
11 | 8, 9, 10 | caovcld 5654 | . 2 |
12 | 4 | caovassg 5659 | . . . . 5 |
13 | 12 | adantlr 446 | . . . 4 |
14 | 13, 9, 10, 11 | caovassd 5660 | . . 3 |
15 | grprinvd.e | . . . . . 6 | |
16 | 15 | oveq1d 5527 | . . . . 5 |
17 | 13, 10, 9, 10 | caovassd 5660 | . . . . 5 |
18 | 3 | ralrimiva 2392 | . . . . . . . 8 |
19 | oveq2 5520 | . . . . . . . . . 10 | |
20 | id 19 | . . . . . . . . . 10 | |
21 | 19, 20 | eqeq12d 2054 | . . . . . . . . 9 |
22 | 21 | cbvralv 2533 | . . . . . . . 8 |
23 | 18, 22 | sylib 127 | . . . . . . 7 |
24 | 23 | adantr 261 | . . . . . 6 |
25 | oveq2 5520 | . . . . . . . 8 | |
26 | id 19 | . . . . . . . 8 | |
27 | 25, 26 | eqeq12d 2054 | . . . . . . 7 |
28 | 27 | rspcv 2652 | . . . . . 6 |
29 | 10, 24, 28 | sylc 56 | . . . . 5 |
30 | 16, 17, 29 | 3eqtr3d 2080 | . . . 4 |
31 | 30 | oveq2d 5528 | . . 3 |
32 | 14, 31 | eqtrd 2072 | . 2 |
33 | 1, 2, 3, 4, 5, 11, 32 | grprinvlem 5695 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wral 2306 wrex 2307 (class class class)co 5512 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: grpridd 5697 |
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