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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 11327 | . 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 1462 | 1 โข (๐ด ยท (๐ต ยท ๐ถ)) = (๐ต ยท (๐ด ยท ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7362 โcc 11056 ยท cmul 11063 |
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 11122 ax-mulass 11124 |
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 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-ov 7365 |
This theorem is referenced by: decmul10add 12694 faclbnd4lem1 14200 bpoly3 15948 decsplit 16962 root1eq1 26124 cxpeq 26126 1cubrlem 26207 efiatan2 26283 2efiatan 26284 tanatan 26285 log2ublem2 26313 log2ublem3 26314 bposlem8 26655 ax5seglem7 27926 ip1ilem 29810 ipasslem10 29823 polid2i 30141 3exp4mod41 45882 |
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