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Theorem divsubdirap 8888
Description: Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
Assertion
Ref Expression
divsubdirap |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )
Proof of Theorem divsubdirap
StepHypRef Expression
1 negcl 8379 . . . 4 |- ( B e. CC -> -u B e. CC )
2 divdirap 8877 . . . 4 |- ( ( A e. CC /\ -u B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A / C ) + ( -u B / C ) ) )
31, 2syl3an2 1307 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A / C ) + ( -u B / C ) ) )
4 negsub 8427 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
54oveq1d 6033 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) / C ) = ( ( A - B ) / C ) )
653adant3 1043 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A - B ) / C ) )
73, 6eqtr3d 2266 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + ( -u B / C ) ) = ( ( A - B ) / C ) )
8 divnegap 8886 . . . . . 6 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> -u ( B / C ) = ( -u B / C ) )
983expb 1230 . . . . 5 |- ( ( B e. CC /\ ( C e. CC /\ C # 0 ) ) -> -u ( B / C ) = ( -u B / C ) )
1093adant1 1041 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> -u ( B / C ) = ( -u B / C ) )
1110oveq2d 6034 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + -u ( B / C ) ) = ( ( A / C ) + ( -u B / C ) ) )
12 divclap 8858 . . . . . 6 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) e. CC )
13123expb 1230 . . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) e. CC )
14133adant2 1042 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) e. CC )
15 divclap 8858 . . . . . 6 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> ( B / C ) e. CC )
16153expb 1230 . . . . 5 |- ( ( B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) e. CC )
17163adant1 1041 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) e. CC )
1814, 17negsubd 8496 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + -u ( B / C ) ) = ( ( A / C ) - ( B / C ) ) )
1911, 18eqtr3d 2266 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + ( -u B / C ) ) = ( ( A / C ) - ( B / C ) ) )
207, 19eqtr3d 2266 1 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6018   CCcc 8030   0cc0 8032   + caddc 8035   - cmin 8350   -ucneg 8351   # cap 8761   / cdiv 8852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853
This theorem is referenced by:  divsubdirapd  9010  1mhlfehlf  9362  halfpm6th  9364  halfaddsub  9378  zeo  9585  mulsubdivbinom2ap  10974  cos2bnd  12326  sinq12gt0  15560  sincos6thpi  15572
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