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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 11840 | . 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 1375 | 1 โข (๐ โ ((๐ด ยท ๐ต) / ๐ถ) = ((๐ด / ๐ถ) ยท ๐ต)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โ wne 2940 (class class class)co 7361 โcc 11057 0cc0 11059 ยท cmul 11064 / cdiv 11820 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 |
This theorem is referenced by: bcpasc 14230 abslem2 15233 geolim 15763 bpolydiflem 15945 efaddlem 15983 eftlub 15999 bitsinv1lem 16329 pjthlem1 24824 itg2monolem3 25140 dvmulbr 25326 dvrecg 25360 dvmptdiv 25361 dvtaylp 25752 itgulm 25790 tanregt0 25918 logtayl2 26040 cxpeq 26133 heron 26211 dcubic2 26217 cubic2 26221 dquartlem1 26224 dquartlem2 26225 dquart 26226 quart1lem 26228 quart1 26229 dvatan 26308 atantayl 26310 jensenlem2 26360 lgamgulmlem2 26402 lgamgulmlem3 26403 ftalem2 26446 bclbnd 26651 bposlem9 26663 lgseisenlem4 26749 lgsquadlem1 26751 lgsquadlem2 26752 dchrvmasumlem1 26866 mulog2sumlem2 26906 2vmadivsumlem 26911 selberg3lem1 26928 selberg4lem1 26931 selberg4 26932 selberg3r 26940 pntrlog2bndlem4 26951 pntrlog2bndlem5 26952 pntibndlem2 26962 pntlemo 26978 brbtwn2 27903 colinearalg 27908 axsegconlem10 27924 axpaschlem 27938 axcontlem8 27969 pjhthlem1 30382 sinccvglem 34324 knoppndvlem14 35041 bj-bary1lem 35831 dvtan 36178 lcmineqlem10 40545 aks4d1p1p7 40581 binomcxplemnotnn0 42728 dvnprodlem2 44278 itgsinexp 44286 stirlinglem3 44407 stirlinglem4 44408 dirkertrigeqlem3 44431 fourierdlem95 44532 eenglngeehlnmlem1 46913 eenglngeehlnmlem2 46914 |
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