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Mirrors > Home > MPE Home > Th. List > divcan5d | Structured version Visualization version GIF version |
Description: Cancellation of common factor in a ratio. (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 |
---|---|
divcan5d | โข (๐ โ ((๐ถ ยท ๐ด) / (๐ถ ยท ๐ต)) = (๐ด / 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.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 | divcan5 11956 | . 2 โข ((๐ด โ โ โง (๐ต โ โ โง ๐ต โ 0) โง (๐ถ โ โ โง ๐ถ โ 0)) โ ((๐ถ ยท ๐ด) / (๐ถ ยท ๐ต)) = (๐ด / ๐ต)) | |
7 | 1, 2, 3, 4, 5, 6 | syl122anc 1376 | 1 โข (๐ โ ((๐ถ ยท ๐ด) / (๐ถ ยท ๐ต)) = (๐ด / ๐ต)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โ wne 2937 (class class class)co 7426 โcc 11146 0cc0 11148 ยท cmul 11153 / cdiv 11911 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 |
This theorem is referenced by: divcan5rd 12057 discr 14244 bcm1k 14316 bcval5 14319 trireciplem 15850 tanval3 16120 tanadd 16153 flodddiv4 16399 bitsinv1lem 16425 lcmgcdlem 16586 pjthlem1 25393 tanarg 26581 advlogexp 26617 angcan 26762 isosctrlem2 26779 mcubic 26807 cubic2 26808 dquart 26813 2efiatan 26878 dvatan 26895 cxp2limlem 26936 chebbnd1lem3 27432 pntrsumo1 27526 pnt 27575 pjhthlem1 31229 subfaclim 34839 faclimlem1 35378 areacirclem1 37222 cxpi11d 41963 binomcxplemwb 43834 dirkertrigeqlem1 45533 dirkercncflem1 45538 itsclc0yqsol 47933 itscnhlc0xyqsol 47934 itschlc0xyqsol1 47935 itschlc0xyqsol 47936 itsclc0xyqsolr 47938 |
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