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Mirrors > Home > MPE Home > Th. List > divdiv1d | Structured version Visualization version GIF version |
Description: Division into a fraction. (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) |
divmuld.4 | โข (๐ โ ๐ต โ 0) |
divdiv23d.5 | โข (๐ โ ๐ถ โ 0) |
Ref | Expression |
---|---|
divdiv1d | โข (๐ โ ((๐ด / ๐ต) / ๐ถ) = (๐ด / (๐ต ยท ๐ถ))) |
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.4 | . 2 โข (๐ โ ๐ต โ 0) | |
4 | divmuld.3 | . 2 โข (๐ โ ๐ถ โ โ) | |
5 | divdiv23d.5 | . 2 โข (๐ โ ๐ถ โ 0) | |
6 | divdiv1 11927 | . 2 โข ((๐ด โ โ โง (๐ต โ โ โง ๐ต โ 0) โง (๐ถ โ โ โง ๐ถ โ 0)) โ ((๐ด / ๐ต) / ๐ถ) = (๐ด / (๐ต ยท ๐ถ))) | |
7 | 1, 2, 3, 4, 5, 6 | syl122anc 1379 | 1 โข (๐ โ ((๐ด / ๐ต) / ๐ถ) = (๐ด / (๐ต ยท ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 โ wne 2940 (class class class)co 7411 โcc 11110 0cc0 11112 ยท cmul 11117 / cdiv 11873 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 |
This theorem is referenced by: discr 14205 hashf1 14420 bcfallfac 15990 eftlub 16054 tanval2 16078 sinhval 16099 sqrt2irrlem 16193 bitsp1 16374 4sqlem7 16879 4sqlem10 16882 uniioombl 25113 dvrec 25479 dvsincos 25505 dvcvx 25544 taylthlem2 25893 mcubic 26359 cubic2 26360 quart1lem 26367 quart1 26368 log2cnv 26456 log2tlbnd 26457 birthdaylem2 26464 efrlim 26481 bcmono 26787 m1lgs 26898 chto1lb 26988 vmalogdivsum2 27048 selberg3lem1 27067 selberg4lem1 27070 selberg4 27071 selberg34r 27081 pntrlog2bndlem2 27088 pntrlog2bndlem4 27090 pntpbnd2 27097 pntibndlem2 27101 pntlemg 27108 nnproddivdvdsd 40952 dvrelogpow2b 41019 aks4d1p1p7 41025 irrapxlem5 41646 divdiv3d 44148 mccllem 44392 clim1fr1 44396 sinaover2ne0 44663 dvnprodlem2 44742 wallispi2lem1 44866 stirlinglem3 44871 stirlinglem4 44872 stirlinglem7 44875 stirlinglem15 44883 dirker2re 44887 dirkerdenne0 44888 dirkertrigeqlem2 44894 dirkertrigeqlem3 44895 dirkertrigeq 44896 dirkercncflem1 44898 dirkercncflem2 44899 dirkercncflem4 44901 fourierdlem56 44957 fourierdlem66 44967 sqwvfourb 45024 fouriersw 45026 itscnhlc0xyqsol 47529 |
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