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Mirrors > Home > MPE Home > Th. List > divcan3d | Structured version Visualization version GIF version |
Description: A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divcld.3 | ⊢ (𝜑 → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
divcan3d | ⊢ (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divcld.3 | . 2 ⊢ (𝜑 → 𝐵 ≠ 0) | |
4 | divcan3 11043 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | |
5 | 1, 2, 3, 4 | syl3anc 1494 | 1 ⊢ (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 (class class class)co 6910 ℂcc 10257 0cc0 10259 · cmul 10264 / cdiv 11016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 |
This theorem is referenced by: prodgt0 11205 mulge0b 11230 ltdivmul 11235 ledivmul 11236 zneo 11795 2tnp1ge0ge0 12932 quoremz 12956 quoremnn0ALT 12958 moddiffl 12983 zesq 13288 discr 13302 bcn1 13400 crre 14238 abslem2 14463 fallfacval4 15153 sinhval 15263 eirrlem 15313 sqrt2irrlem 15358 ltoddhalfle 15466 flodddiv4 15517 bitsp1e 15534 bitsp1o 15535 iserodd 15918 fldivp1 15979 4sqlem17 16043 gexexlem 18615 abv1z 19195 gzrngunit 20179 cphipval2 23416 ovolunlem1a 23669 itg1mulc 23877 dvrec 24124 elqaalem3 24482 eff1olem 24701 logf1o2 24802 isosctrlem2 24966 heron 24985 dcubic2 24991 mcubic 24994 cubic2 24995 dquartlem1 24998 dquartlem2 24999 dquart 25000 cosasin 25051 efiatan2 25064 tanatan 25066 dvatan 25082 atantayl3 25086 jensen 25135 basellem3 25229 basellem5 25231 basellem8 25234 logfacrlim 25369 perfectlem2 25375 lgsquadlem1 25525 lgsquadlem2 25526 2lgslem1c 25538 2lgslem3a 25541 dchrvmasumlem1 25604 mudivsum 25639 vmalogdivsum2 25647 logsqvma 25651 selberglem2 25655 selberglem3 25656 selberg 25657 selbergr 25677 selberg3r 25678 selberg4r 25679 selberg34r 25680 pntsval2 25685 pntpbnd1a 25694 pntibndlem2 25700 axsegconlem9 26231 cdj1i 29843 subfacval2 31711 circum 32108 knoppndvlem2 33031 knoppndvlem9 33038 areacirclem1 34038 areacirclem4 34041 hashnzfzclim 39356 dmmcand 40319 sumnnodd 40651 sinmulcos 40865 itgsinexp 40959 itgcoscmulx 40973 itgsincmulx 40978 stirlinglem7 41085 dirkertrigeqlem3 41105 dirkeritg 41107 dirkercncflem2 41109 fourierdlem79 41190 fourierdlem83 41194 fourierdlem95 41206 fouriercnp 41231 fourierswlem 41235 etransclem24 41263 etransclem41 41280 sfprmdvdsmersenne 42364 dfodd6 42394 dfeven4 42395 perfectALTVlem2 42475 line2 43314 itsclc0lem5 43322 sinhpcosh 43389 |
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