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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 8116 |
. 2
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5 | 1, 2, 3, 4 | syl3anc 1249 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-mulcom 7941 ax-mulass 7943 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5898 |
This theorem is referenced by: conjmulap 8715 modqmul1 10407 binom3 10668 bernneq 10671 bcm1k 10771 bcp1n 10772 resqrexlemcalc1 11054 resqrexlemnm 11058 reccn2ap 11352 binomlem 11522 tanaddap 11778 eirraplem 11815 dvds2ln 11862 divgcdcoprm0 12132 modprm0 12285 binom4 14849 |
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