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Mirrors > Home > ILE Home > Th. List > isabl2 | Unicode version |
Description: The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
iscmn.b | |
iscmn.p |
Ref | Expression |
---|---|
isabl2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isabl 12888 | . 2 CMnd | |
2 | grpmnd 12745 | . . . 4 | |
3 | iscmn.b | . . . . . 6 | |
4 | iscmn.p | . . . . . 6 | |
5 | 3, 4 | iscmn 12892 | . . . . 5 CMnd |
6 | 5 | baib 919 | . . . 4 CMnd |
7 | 2, 6 | syl 14 | . . 3 CMnd |
8 | 7 | pm5.32i 454 | . 2 CMnd |
9 | 1, 8 | bitri 184 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 104 wb 105 wceq 1353 wcel 2146 wral 2453 cfv 5208 (class class class)co 5865 cbs 12428 cplusg 12492 cmnd 12682 cgrp 12738 CMndccmn 12884 cabl 12885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-ext 2157 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-un 3131 df-in 3133 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-iota 5170 df-fv 5216 df-ov 5868 df-grp 12741 df-cmn 12886 df-abl 12887 |
This theorem is referenced by: isabli 12899 |
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