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Mirrors > Home > MPE Home > Th. List > mul12i | Structured version Visualization version GIF version |
Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
mul.1 | โข ๐ด โ โ |
mul.2 | โข ๐ต โ โ |
mul.3 | โข ๐ถ โ โ |
Ref | Expression |
---|---|
mul12i | โข (๐ด ยท (๐ต ยท ๐ถ)) = (๐ต ยท (๐ด ยท ๐ถ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.1 | . 2 โข ๐ด โ โ | |
2 | mul.2 | . 2 โข ๐ต โ โ | |
3 | mul.3 | . 2 โข ๐ถ โ โ | |
4 | mul12 11375 | . 2 โข ((๐ด โ โ โง ๐ต โ โ โง ๐ถ โ โ) โ (๐ด ยท (๐ต ยท ๐ถ)) = (๐ต ยท (๐ด ยท ๐ถ))) | |
5 | 1, 2, 3, 4 | mp3an 1461 | 1 โข (๐ด ยท (๐ต ยท ๐ถ)) = (๐ต ยท (๐ด ยท ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 โ wcel 2106 (class class class)co 7405 โcc 11104 ยท cmul 11111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-mulcom 11170 ax-mulass 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7408 |
This theorem is referenced by: decmul10add 12742 faclbnd4lem1 14249 bpoly3 15998 decsplit 17012 root1eq1 26252 cxpeq 26254 1cubrlem 26335 efiatan2 26411 2efiatan 26412 tanatan 26413 log2ublem2 26441 log2ublem3 26442 bposlem8 26783 ax5seglem7 28182 ip1ilem 30066 ipasslem10 30079 polid2i 30397 3exp4mod41 46270 |
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