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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 | |
addcomd.2 | |
mul12d.3 |
Ref | Expression |
---|---|
mul32d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muld.1 | . 2 | |
2 | addcomd.2 | . 2 | |
3 | mul12d.3 | . 2 | |
4 | mul32 8038 | . 2 | |
5 | 1, 2, 3, 4 | syl3anc 1233 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 (class class class)co 5851 cc 7761 cmul 7768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-mulcom 7864 ax-mulass 7866 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5854 |
This theorem is referenced by: conjmulap 8635 modqmul1 10322 binom3 10582 bernneq 10585 bcm1k 10683 bcp1n 10684 resqrexlemcalc1 10967 resqrexlemnm 10971 reccn2ap 11265 binomlem 11435 tanaddap 11691 eirraplem 11728 dvds2ln 11775 divgcdcoprm0 12044 modprm0 12197 binom4 13652 |
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