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Mirrors > Home > MPE Home > Th. List > div23d | Structured version Visualization version GIF version |
Description: A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | โข (๐ โ ๐ด โ โ) |
divcld.2 | โข (๐ โ ๐ต โ โ) |
divmuld.3 | โข (๐ โ ๐ถ โ โ) |
divassd.4 | โข (๐ โ ๐ถ โ 0) |
Ref | Expression |
---|---|
div23d | โข (๐ โ ((๐ด ยท ๐ต) / ๐ถ) = ((๐ด / ๐ถ) ยท ๐ต)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 โข (๐ โ ๐ด โ โ) | |
2 | divcld.2 | . 2 โข (๐ โ ๐ต โ โ) | |
3 | divmuld.3 | . 2 โข (๐ โ ๐ถ โ โ) | |
4 | divassd.4 | . 2 โข (๐ โ ๐ถ โ 0) | |
5 | div23 11915 | . 2 โข ((๐ด โ โ โง ๐ต โ โ โง (๐ถ โ โ โง ๐ถ โ 0)) โ ((๐ด ยท ๐ต) / ๐ถ) = ((๐ด / ๐ถ) ยท ๐ต)) | |
6 | 1, 2, 3, 4, 5 | syl112anc 1372 | 1 โข (๐ โ ((๐ด ยท ๐ต) / ๐ถ) = ((๐ด / ๐ถ) ยท ๐ต)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 โ wne 2936 (class class class)co 7414 โcc 11130 0cc0 11132 ยท cmul 11137 / cdiv 11895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 |
This theorem is referenced by: bcpasc 14306 abslem2 15312 geolim 15842 bpolydiflem 16024 efaddlem 16063 eftlub 16079 bitsinv1lem 16409 pjthlem1 25358 itg2monolem3 25675 dvmulbr 25862 dvmulbrOLD 25863 dvrecg 25898 dvmptdiv 25899 dvtaylp 26298 itgulm 26337 tanregt0 26466 logtayl2 26589 cxpeq 26685 heron 26763 dcubic2 26769 cubic2 26773 dquartlem1 26776 dquartlem2 26777 dquart 26778 quart1lem 26780 quart1 26781 dvatan 26860 atantayl 26862 jensenlem2 26913 lgamgulmlem2 26955 lgamgulmlem3 26956 ftalem2 26999 bclbnd 27206 bposlem9 27218 lgseisenlem4 27304 lgsquadlem1 27306 lgsquadlem2 27307 dchrvmasumlem1 27421 mulog2sumlem2 27461 2vmadivsumlem 27466 selberg3lem1 27483 selberg4lem1 27486 selberg4 27487 selberg3r 27495 pntrlog2bndlem4 27506 pntrlog2bndlem5 27507 pntibndlem2 27517 pntlemo 27533 brbtwn2 28709 colinearalg 28714 axsegconlem10 28730 axpaschlem 28744 axcontlem8 28775 pjhthlem1 31194 sinccvglem 35270 knoppndvlem14 35994 bj-bary1lem 36783 dvtan 37137 lcmineqlem10 41503 aks4d1p1p7 41539 cxpi11d 41908 binomcxplemnotnn0 43787 dvnprodlem2 45329 itgsinexp 45337 stirlinglem3 45458 stirlinglem4 45459 dirkertrigeqlem3 45482 fourierdlem95 45583 eenglngeehlnmlem1 47804 eenglngeehlnmlem2 47805 |
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