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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 1199 | . . . . 5 |
7 | 6 | caovclg 6005 | . . . 4 |
8 | 7 | adantlr 474 | . . 3 |
9 | grprinvd.x | . . 3 | |
10 | grprinvd.n | . . 3 | |
11 | 8, 9, 10 | caovcld 6006 | . 2 |
12 | 4 | caovassg 6011 | . . . . 5 |
13 | 12 | adantlr 474 | . . . 4 |
14 | 13, 9, 10, 11 | caovassd 6012 | . . 3 |
15 | grprinvd.e | . . . . . 6 | |
16 | 15 | oveq1d 5868 | . . . . 5 |
17 | 13, 10, 9, 10 | caovassd 6012 | . . . . 5 |
18 | oveq2 5861 | . . . . . . 7 | |
19 | id 19 | . . . . . . 7 | |
20 | 18, 19 | eqeq12d 2185 | . . . . . 6 |
21 | 3 | ralrimiva 2543 | . . . . . . . 8 |
22 | oveq2 5861 | . . . . . . . . . 10 | |
23 | id 19 | . . . . . . . . . 10 | |
24 | 22, 23 | eqeq12d 2185 | . . . . . . . . 9 |
25 | 24 | cbvralvw 2700 | . . . . . . . 8 |
26 | 21, 25 | sylib 121 | . . . . . . 7 |
27 | 26 | adantr 274 | . . . . . 6 |
28 | 20, 27, 10 | rspcdva 2839 | . . . . 5 |
29 | 16, 17, 28 | 3eqtr3d 2211 | . . . 4 |
30 | 29 | oveq2d 5869 | . . 3 |
31 | 14, 30 | eqtrd 2203 | . 2 |
32 | 1, 2, 3, 4, 5, 11, 31 | grprinvlem 12639 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 wral 2448 wrex 2449 (class class class)co 5853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: grpridd 12641 grprcan 12740 grprinv 12753 |
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