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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 | |
addcomd.2 | |
mul12d.3 |
Ref | Expression |
---|---|
mul12d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muld.1 | . 2 | |
2 | addcomd.2 | . 2 | |
3 | mul12d.3 | . 2 | |
4 | mul12 8004 | . 2 | |
5 | 1, 2, 3, 4 | syl3anc 1220 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 wcel 2128 (class class class)co 5824 cc 7730 cmul 7737 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-mulcom 7833 ax-mulass 7835 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-un 3106 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-iota 5135 df-fv 5178 df-ov 5827 |
This theorem is referenced by: mulreim 8479 divrecap 8561 remullem 10771 cvgratnnlemnexp 11421 cvgratnnlemmn 11422 tanval3ap 11611 sinadd 11633 dvdscmulr 11715 bezoutlemnewy 11879 dvdsmulgcd 11908 lcmgcdlem 11953 cncongr1 11979 prmdiv 12109 tangtx 13159 |
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