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Mirrors > Home > MPE Home > Th. List > mul32d | Structured version Visualization version GIF 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 | โข (๐ โ ๐ด โ โ) |
addcomd.2 | โข (๐ โ ๐ต โ โ) |
addcand.3 | โข (๐ โ ๐ถ โ โ) |
Ref | Expression |
---|---|
mul32d | โข (๐ โ ((๐ด ยท ๐ต) ยท ๐ถ) = ((๐ด ยท ๐ถ) ยท ๐ต)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muld.1 | . 2 โข (๐ โ ๐ด โ โ) | |
2 | addcomd.2 | . 2 โข (๐ โ ๐ต โ โ) | |
3 | addcand.3 | . 2 โข (๐ โ ๐ถ โ โ) | |
4 | mul32 11322 | . 2 โข ((๐ด โ โ โง ๐ต โ โ โง ๐ถ โ โ) โ ((๐ด ยท ๐ต) ยท ๐ถ) = ((๐ด ยท ๐ถ) ยท ๐ต)) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | 1 โข (๐ โ ((๐ด ยท ๐ต) ยท ๐ถ) = ((๐ด ยท ๐ถ) ยท ๐ต)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 (class class class)co 7358 โcc 11050 ยท cmul 11057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-mulcom 11116 ax-mulass 11118 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 |
This theorem is referenced by: conjmul 11873 modmul1 13830 binom3 14128 bernneq 14133 expmulnbnd 14139 discr 14144 bcm1k 14216 bcp1n 14217 reccn2 15480 binomlem 15715 binomfallfaclem2 15924 tanadd 16050 eirrlem 16087 dvds2ln 16172 bezoutlem4 16424 divgcdcoprm0 16542 modprm0 16678 nrginvrcnlem 24058 tcphcphlem2 24603 csbren 24766 radcnvlem1 25775 tanarg 25977 cxpeq 26113 quad2 26192 binom4 26203 dquartlem2 26205 dquart 26206 quart1lem 26208 dvatan 26288 log2cnv 26297 basellem8 26440 bcmono 26628 gausslemma2d 26725 lgsquadlem1 26731 2lgslem3b 26748 2lgslem3c 26749 2lgslem3d 26750 rplogsumlem1 26835 dchrisumlem2 26841 chpdifbndlem1 26904 selberg3lem1 26908 selberg4 26912 selberg3r 26920 pntrlog2bndlem2 26929 pntrlog2bndlem3 26930 pntrlog2bndlem5 26932 pntlemf 26956 pntlemo 26958 ostth2lem1 26969 ostth2lem3 26986 logdivsqrle 33266 circum 34265 lcmineqlem8 40496 lcmineqlem12 40500 flt4lem5f 40998 jm2.25 41326 jm2.27c 41334 binomcxplemnotnn0 42643 dvasinbx 44168 stirlinglem3 44324 dirkercncflem2 44352 cevathlem1 45115 itschlc0yqe 46853 |
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