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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 8048 | . 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 5853 cc 7772 cmul 7779 |
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 7875 ax-mulass 7877 |
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 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: mulreim 8523 divrecap 8605 remullem 10835 cvgratnnlemnexp 11487 cvgratnnlemmn 11488 tanval3ap 11677 sinadd 11699 dvdscmulr 11782 bezoutlemnewy 11951 dvdsmulgcd 11980 lcmgcdlem 12031 cncongr1 12057 prmdiv 12189 tangtx 13553 2sqlem4 13748 |
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