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Mirrors > Home > ILE Home > Th. List > mul32 | Unicode version |
Description: Commutative/associative law. (Contributed by NM, 8-Oct-1999.) |
Ref | Expression |
---|---|
mul32 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcom 7666 |
. . . 4
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2 | 1 | oveq2d 5742 |
. . 3
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3 | 2 | 3adant1 980 |
. 2
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4 | mulass 7668 |
. 2
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5 | mulass 7668 |
. . 3
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6 | 5 | 3com23 1168 |
. 2
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7 | 3, 4, 6 | 3eqtr4d 2155 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-mulcom 7639 ax-mulass 7641 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-rex 2394 df-v 2657 df-un 3039 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-iota 5044 df-fv 5087 df-ov 5729 |
This theorem is referenced by: mul4 7810 mul32i 7825 mul32d 7831 muldvds1 11359 |
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