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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 5781 | . . . . . 6 | |
3 | 2 | eqeq1d 2148 | . . . . 5 |
4 | 3 | cbvrexv 2655 | . . . 4 |
5 | 1, 4 | sylib 121 | . . 3 |
6 | grprinvlem.a | . . . . . . . 8 | |
7 | 6 | caovassg 5929 | . . . . . . 7 |
8 | 7 | adantlr 468 | . . . . . 6 |
9 | simprl 520 | . . . . . 6 | |
10 | simprrl 528 | . . . . . 6 | |
11 | 8, 9, 10, 9 | caovassd 5930 | . . . . 5 |
12 | grprinvlem.c | . . . . . . 7 | |
13 | grprinvlem.o | . . . . . . 7 | |
14 | grprinvlem.i | . . . . . . 7 | |
15 | simprrr 529 | . . . . . . 7 | |
16 | 12, 13, 14, 6, 1, 9, 10, 15 | grprinvd 5966 | . . . . . 6 |
17 | 16 | oveq1d 5789 | . . . . 5 |
18 | 15 | oveq2d 5790 | . . . . 5 |
19 | 11, 17, 18 | 3eqtr3d 2180 | . . . 4 |
20 | 19 | anassrs 397 | . . 3 |
21 | 5, 20 | rexlimddv 2554 | . 2 |
22 | 21, 14 | eqtr3d 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wrex 2417 (class class class)co 5774 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: (None) |
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