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Mirrors > Home > ILE Home > Th. List > mul12d | Unicode version |
Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
muld.1 |
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addcomd.2 |
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mul12d.3 |
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Ref | Expression |
---|---|
mul12d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muld.1 |
. 2
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2 | addcomd.2 |
. 2
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3 | mul12d.3 |
. 2
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4 | mul12 7513 |
. 2
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5 | 1, 2, 3, 4 | syl3anc 1170 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-mulcom 7348 ax-mulass 7350 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rex 2359 df-v 2614 df-un 2988 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-iota 4933 df-fv 4976 df-ov 5593 |
This theorem is referenced by: mulreim 7980 divrecap 8052 remullem 10131 dvdscmulr 10604 bezoutlemnewy 10764 dvdsmulgcd 10793 lcmgcdlem 10838 cncongr1 10864 |
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