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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 11889 | . 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 1371 | 1 โข (๐ โ ((๐ด ยท ๐ต) / ๐ถ) = ((๐ด / ๐ถ) ยท ๐ต)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โ wne 2932 (class class class)co 7402 โcc 11105 0cc0 11107 ยท cmul 11112 / cdiv 11869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 |
This theorem is referenced by: bcpasc 14279 abslem2 15284 geolim 15814 bpolydiflem 15996 efaddlem 16035 eftlub 16051 bitsinv1lem 16381 pjthlem1 25289 itg2monolem3 25606 dvmulbr 25793 dvmulbrOLD 25794 dvrecg 25829 dvmptdiv 25830 dvtaylp 26225 itgulm 26263 tanregt0 26392 logtayl2 26515 cxpeq 26611 heron 26689 dcubic2 26695 cubic2 26699 dquartlem1 26702 dquartlem2 26703 dquart 26704 quart1lem 26706 quart1 26707 dvatan 26786 atantayl 26788 jensenlem2 26839 lgamgulmlem2 26881 lgamgulmlem3 26882 ftalem2 26925 bclbnd 27132 bposlem9 27144 lgseisenlem4 27230 lgsquadlem1 27232 lgsquadlem2 27233 dchrvmasumlem1 27347 mulog2sumlem2 27387 2vmadivsumlem 27392 selberg3lem1 27409 selberg4lem1 27412 selberg4 27413 selberg3r 27421 pntrlog2bndlem4 27432 pntrlog2bndlem5 27433 pntibndlem2 27443 pntlemo 27459 brbtwn2 28635 colinearalg 28640 axsegconlem10 28656 axpaschlem 28670 axcontlem8 28701 pjhthlem1 31116 sinccvglem 35148 knoppndvlem14 35892 bj-bary1lem 36682 dvtan 37032 lcmineqlem10 41400 aks4d1p1p7 41436 cxpi11d 41746 binomcxplemnotnn0 43629 dvnprodlem2 45173 itgsinexp 45181 stirlinglem3 45302 stirlinglem4 45303 dirkertrigeqlem3 45326 fourierdlem95 45427 eenglngeehlnmlem1 47636 eenglngeehlnmlem2 47637 |
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