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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 11930 | . 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 1378 | 1 โข (๐ โ ((๐ด / ๐ต) / ๐ถ) = (๐ด / (๐ต ยท ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1540 โ wcel 2105 โ wne 2939 (class class class)co 7412 โcc 11112 0cc0 11114 ยท cmul 11119 / cdiv 11876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 |
This theorem is referenced by: discr 14208 hashf1 14423 bcfallfac 15993 eftlub 16057 tanval2 16081 sinhval 16102 sqrt2irrlem 16196 bitsp1 16377 4sqlem7 16882 4sqlem10 16885 uniioombl 25339 dvrec 25708 dvsincos 25734 dvcvx 25773 taylthlem2 26123 mcubic 26589 cubic2 26590 quart1lem 26597 quart1 26598 log2cnv 26686 log2tlbnd 26687 birthdaylem2 26694 efrlim 26711 bcmono 27017 m1lgs 27128 chto1lb 27218 vmalogdivsum2 27278 selberg3lem1 27297 selberg4lem1 27300 selberg4 27301 selberg34r 27311 pntrlog2bndlem2 27318 pntrlog2bndlem4 27320 pntpbnd2 27327 pntibndlem2 27331 pntlemg 27338 nnproddivdvdsd 41173 dvrelogpow2b 41240 aks4d1p1p7 41246 irrapxlem5 41867 divdiv3d 44368 mccllem 44612 clim1fr1 44616 sinaover2ne0 44883 dvnprodlem2 44962 wallispi2lem1 45086 stirlinglem3 45091 stirlinglem4 45092 stirlinglem7 45095 stirlinglem15 45103 dirker2re 45107 dirkerdenne0 45108 dirkertrigeqlem2 45114 dirkertrigeqlem3 45115 dirkertrigeq 45116 dirkercncflem1 45118 dirkercncflem2 45119 dirkercncflem4 45121 fourierdlem56 45177 fourierdlem66 45187 sqwvfourb 45244 fouriersw 45246 itscnhlc0xyqsol 47539 |
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