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Mirrors > Home > ILE Home > Th. List > cmncom | Unicode version |
Description: A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
ablcom.b |
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ablcom.p |
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Ref | Expression |
---|---|
cmncom |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablcom.b |
. . . . . 6
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2 | ablcom.p |
. . . . . 6
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3 | 1, 2 | iscmn 13027 |
. . . . 5
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4 | 3 | simprbi 275 |
. . . 4
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5 | rsp2 2527 |
. . . . 5
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6 | 5 | imp 124 |
. . . 4
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7 | 4, 6 | sylan 283 |
. . 3
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8 | 7 | caovcomg 6027 |
. 2
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9 | 8 | 3impb 1199 |
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 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-iota 5177 df-fv 5223 df-ov 5875 df-cmn 13021 |
This theorem is referenced by: ablcom 13037 cmn32 13038 cmn4 13039 cmn12 13040 rinvmod 13043 subcmnd 13060 srgcom 13097 crngcom 13128 |
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