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Mirrors > Home > ILE Home > Th. List > mul31 | Unicode version |
Description: Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
mul31 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcom 7469 |
. . . 4
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2 | 1 | oveq2d 5668 |
. . 3
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3 | 2 | 3adant1 961 |
. 2
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4 | mulass 7471 |
. 2
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5 | mulcl 7467 |
. . . . 5
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6 | 5 | ancoms 264 |
. . . 4
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7 | 6 | 3adant1 961 |
. . 3
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8 | simp1 943 |
. . 3
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9 | 7, 8 | mulcomd 7507 |
. 2
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10 | 3, 4, 9 | 3eqtr4d 2130 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-mulcl 7441 ax-mulcom 7444 ax-mulass 7446 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rex 2365 df-v 2621 df-un 3003 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-iota 4980 df-fv 5023 df-ov 5655 |
This theorem is referenced by: mul31d 7634 |
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