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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 5831 | . . . . . 6 | |
3 | 2 | eqeq1d 2166 | . . . . 5 |
4 | 3 | cbvrexv 2681 | . . . 4 |
5 | 1, 4 | sylib 121 | . . 3 |
6 | grprinvlem.a | . . . . . . . 8 | |
7 | 6 | caovassg 5979 | . . . . . . 7 |
8 | 7 | adantlr 469 | . . . . . 6 |
9 | simprl 521 | . . . . . 6 | |
10 | simprrl 529 | . . . . . 6 | |
11 | 8, 9, 10, 9 | caovassd 5980 | . . . . 5 |
12 | grprinvlem.c | . . . . . . 7 | |
13 | grprinvlem.o | . . . . . . 7 | |
14 | grprinvlem.i | . . . . . . 7 | |
15 | simprrr 530 | . . . . . . 7 | |
16 | 12, 13, 14, 6, 1, 9, 10, 15 | grprinvd 6016 | . . . . . 6 |
17 | 16 | oveq1d 5839 | . . . . 5 |
18 | 15 | oveq2d 5840 | . . . . 5 |
19 | 11, 17, 18 | 3eqtr3d 2198 | . . . 4 |
20 | 19 | anassrs 398 | . . 3 |
21 | 5, 20 | rexlimddv 2579 | . 2 |
22 | 21, 14 | eqtr3d 2192 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 wrex 2436 (class class class)co 5824 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-iota 5135 df-fv 5178 df-ov 5827 |
This theorem is referenced by: (None) |
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