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Mirrors > Home > ILE Home > Th. List > grpridd | Unicode version |
Description: Deduce right identity 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 |
Ref | Expression |
---|---|
grpridd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grprinvlem.n | . . . 4 | |
2 | oveq1 5519 | . . . . . 6 | |
3 | 2 | eqeq1d 2048 | . . . . 5 |
4 | 3 | cbvrexv 2534 | . . . 4 |
5 | 1, 4 | sylib 127 | . . 3 |
6 | grprinvlem.a | . . . . . . . 8 | |
7 | 6 | caovassg 5659 | . . . . . . 7 |
8 | 7 | adantlr 446 | . . . . . 6 |
9 | simprl 483 | . . . . . 6 | |
10 | simprrl 491 | . . . . . 6 | |
11 | 8, 9, 10, 9 | caovassd 5660 | . . . . 5 |
12 | grprinvlem.c | . . . . . . 7 | |
13 | grprinvlem.o | . . . . . . 7 | |
14 | grprinvlem.i | . . . . . . 7 | |
15 | simprrr 492 | . . . . . . 7 | |
16 | 12, 13, 14, 6, 1, 9, 10, 15 | grprinvd 5696 | . . . . . 6 |
17 | 16 | oveq1d 5527 | . . . . 5 |
18 | 15 | oveq2d 5528 | . . . . 5 |
19 | 11, 17, 18 | 3eqtr3d 2080 | . . . 4 |
20 | 19 | anassrs 380 | . . 3 |
21 | 5, 20 | rexlimddv 2437 | . 2 |
22 | 21, 14 | eqtr3d 2074 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 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: (None) |
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