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Mirrors > Home > ILE Home > Th. List > grpmnd | Unicode version |
Description: A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grpmnd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 |
. . 3
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2 | eqid 2177 |
. . 3
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3 | eqid 2177 |
. . 3
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4 | 1, 2, 3 | isgrp 12773 |
. 2
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5 | 4 | simplbi 274 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-iota 5174 df-fv 5220 df-ov 5872 df-grp 12770 |
This theorem is referenced by: grpcl 12775 grpass 12776 grpideu 12778 grpmndd 12779 grpplusf 12781 grpplusfo 12782 grpsgrp 12791 dfgrp2 12792 grpidcl 12794 grplid 12796 grprid 12797 dfgrp3m 12858 mulgaddcom 12895 mulginvcom 12896 mulgz 12899 mulgneg2 12905 mulgass 12908 isabl2 12924 |
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