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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 5860 | . . . . . 6 | |
3 | 2 | eqeq1d 2179 | . . . . 5 |
4 | 3 | cbvrexvw 2701 | . . . 4 |
5 | 1, 4 | sylib 121 | . . 3 |
6 | grprinvlem.a | . . . . . . . 8 | |
7 | 6 | caovassg 6011 | . . . . . . 7 |
8 | 7 | adantlr 474 | . . . . . 6 |
9 | simprl 526 | . . . . . 6 | |
10 | simprrl 534 | . . . . . 6 | |
11 | 8, 9, 10, 9 | caovassd 6012 | . . . . 5 |
12 | grprinvlem.c | . . . . . . 7 | |
13 | grprinvlem.o | . . . . . . 7 | |
14 | grprinvlem.i | . . . . . . 7 | |
15 | simprrr 535 | . . . . . . 7 | |
16 | 12, 13, 14, 6, 1, 9, 10, 15 | grprinvd 12640 | . . . . . 6 |
17 | 16 | oveq1d 5868 | . . . . 5 |
18 | 15 | oveq2d 5869 | . . . . 5 |
19 | 11, 17, 18 | 3eqtr3d 2211 | . . . 4 |
20 | 19 | anassrs 398 | . . 3 |
21 | 5, 20 | rexlimddv 2592 | . 2 |
22 | 21, 14 | eqtr3d 2205 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 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: isgrpde 12728 |
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