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Mirrors > Home > ILE Home > Th. List > mul32d | Unicode version |
Description: Commutative/associative law that swaps the last 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 |
---|---|
mul32d |
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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 | mul32 7763 |
. 2
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5 | 1, 2, 3, 4 | syl3anc 1184 |
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 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-mulcom 7596 ax-mulass 7598 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rex 2381 df-v 2643 df-un 3025 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-iota 5024 df-fv 5067 df-ov 5709 |
This theorem is referenced by: conjmulap 8350 modqmul1 9991 binom3 10250 bernneq 10253 bcm1k 10347 bcp1n 10348 resqrexlemcalc1 10626 resqrexlemnm 10630 reccn2ap 10921 binomlem 11091 tanaddap 11244 eirraplem 11278 dvds2ln 11321 divgcdcoprm0 11575 |
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