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Mirrors > Home > MPE Home > Th. List > div12d | 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 |
---|---|
div12d | โข (๐ โ (๐ด ยท (๐ต / ๐ถ)) = (๐ต ยท (๐ด / 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.3 | . 2 โข (๐ โ ๐ถ โ โ) | |
4 | divassd.4 | . 2 โข (๐ โ ๐ถ โ 0) | |
5 | div12 11891 | . 2 โข ((๐ด โ โ โง ๐ต โ โ โง (๐ถ โ โ โง ๐ถ โ 0)) โ (๐ด ยท (๐ต / ๐ถ)) = (๐ต ยท (๐ด / ๐ถ))) | |
6 | 1, 2, 3, 4, 5 | syl112anc 1371 | 1 โข (๐ โ (๐ด ยท (๐ต / ๐ถ)) = (๐ต ยท (๐ด / ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โ wne 2932 (class class class)co 7401 โcc 11104 0cc0 11106 ยท cmul 11111 / cdiv 11868 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 |
This theorem is referenced by: bcpasc 14278 geo2sum 15816 bpoly4 16000 eirrlem 16144 dvmptdiv 25828 abelthlem7 26292 lawcoslem1 26663 dcubic1lem 26691 quart1 26704 leibpi 26790 efrlim 26817 efrlimOLD 26818 dchrmusum2 27343 dchrvmasumiflem1 27350 dchrisum0lem2 27367 selberg4lem1 27409 selberg3r 27418 selberg4r 27419 ostth2lem4 27485 subfacval2 34667 knoppndvlem9 35886 geomcau 37117 aks4d1p1p7 41432 stoweidlem11 45212 stirlinglem1 45275 stirlinglem3 45277 |
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