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Theorem divsubdir 10721
Description: Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
Assertion
Ref Expression
divsubdir |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )
Proof of Theorem divsubdir
StepHypRef Expression
1 negcl 10281 . . . 4 |- ( B e. CC -> -u B e. CC )
2 divdir 10710 . . . 4 |- ( ( A e. CC /\ -u B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A / C ) + ( -u B / C ) ) )
31, 2syl3an2 1360 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A / C ) + ( -u B / C ) ) )
4 negsub 10329 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
54oveq1d 6665 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) / C ) = ( ( A - B ) / C ) )
653adant3 1081 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A - B ) / C ) )
73, 6eqtr3d 2658 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / C ) + ( -u B / C ) ) = ( ( A - B ) / C ) )
8 divneg 10719 . . . . . 6 |- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> -u ( B / C ) = ( -u B / C ) )
983expb 1266 . . . . 5 |- ( ( B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> -u ( B / C ) = ( -u B / C ) )
1093adant1 1079 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> -u ( B / C ) = ( -u B / C ) )
1110oveq2d 6666 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / C ) + -u ( B / C ) ) = ( ( A / C ) + ( -u B / C ) ) )
12 divcl 10691 . . . . . 6 |- ( ( A e. CC /\ C e. CC /\ C =/= 0 ) -> ( A / C ) e. CC )
13123expb 1266 . . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( A / C ) e. CC )
14133adant2 1080 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( A / C ) e. CC )
15 divcl 10691 . . . . . 6 |- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( B / C ) e. CC )
16153expb 1266 . . . . 5 |- ( ( B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( B / C ) e. CC )
17163adant1 1079 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( B / C ) e. CC )
1814, 17negsubd 10398 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / C ) + -u ( B / C ) ) = ( ( A / C ) - ( B / C ) ) )
1911, 18eqtr3d 2658 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / C ) + ( -u B / C ) ) = ( ( A / C ) - ( B / C ) ) )
207, 19eqtr3d 2658 1 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   - cmin 10266   -ucneg 10267   / cdiv 10684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685
This theorem is referenced by:  divsubdird  10840  1mhlfehlf  11251  halfpm6th  11253  halfaddsub  11265  zeo  11463  quoremz  12654  quoremnn0ALT  12656  mulsubdivbinom2  13046  facndiv  13075  bpoly3  14789  cos2bnd  14918  rpnnen2lem3  14945  rpnnen2lem11  14953  pythagtriplem15  15534  ovolscalem1  23281  sinq12gt0  24259  sincos6thpi  24267  ang180lem2  24540  log2cnv  24671  log2tlbnd  24672  basellem3  24809  ppiub  24929  logfacrlim  24949  logexprlim  24950  bposlem8  25016  gausslemma2dlem1a  25090  chtppilimlem1  25162  vmadivsum  25171  rplogsumlem2  25174  rpvmasumlem  25176  rplogsum  25216  mulog2sumlem1  25223  selberg2lem  25239  selberg2  25240  selbergr  25257  pntlemr  25291  pntlemj  25292  ballotth  30599  subdivcomb1  31611  subdivcomb2  31612  nndivsub  32456  heiborlem6  33615  areaquad  37802  lhe4.4ex1a  38528  stirlinglem10  40300  divsub1dir  42307
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