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Theorem divcan5 10308
Description: Cancellation of common factor in a ratio. (Contributed by NM, 9-Jan-2006.)
Assertion
Ref Expression
divcan5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  x.  A )  /  ( C  x.  B )
)  =  ( A  /  B ) )

Proof of Theorem divcan5
StepHypRef Expression
1 divid 10296 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( C  /  C
)  =  1 )
21oveq1d 6320 . . 3  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( ( C  /  C )  x.  ( A  /  B ) )  =  ( 1  x.  ( A  /  B
) ) )
323ad2ant3 1028 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  /  C )  x.  ( A  /  B ) )  =  ( 1  x.  ( A  /  B
) ) )
4 simp3l 1033 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
5 simp1 1005 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  A  e.  CC )
6 simp3 1007 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
7 simp2 1006 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
8 divmuldiv 10306 . . 3  |-  ( ( ( C  e.  CC  /\  A  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) ) )  ->  ( ( C  /  C )  x.  ( A  /  B
) )  =  ( ( C  x.  A
)  /  ( C  x.  B ) ) )
94, 5, 6, 7, 8syl22anc 1265 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  /  C )  x.  ( A  /  B ) )  =  ( ( C  x.  A )  / 
( C  x.  B
) ) )
10 divcl 10275 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
11103expb 1206 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  CC )
1211mulid2d 9660 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( 1  x.  ( A  /  B ) )  =  ( A  /  B
) )
13123adant3 1025 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  x.  ( A  /  B ) )  =  ( A  /  B ) )
143, 9, 133eqtr3d 2478 1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  x.  A )  /  ( C  x.  B )
)  =  ( A  /  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625  (class class class)co 6305   CCcc 9536   0cc0 9538   1c1 9539    x. cmul 9543    / cdiv 10268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269
This theorem is referenced by:  divcan7  10315  divadddiv  10321  divcan5d  10408  8th4div3  10833  modmulnn  12111  moddi  12154  reccn2  13638  bpoly3  14089  efif1olem4  23359  ang180lem1  23603  quart1  23647  divsqrtsumlem  23770  basellem1  23870  ppiub  23995  bposlem8  24082  chpchtlim  24180  pnt2  24314  pigt3  31642  dvasin  31732  heiborlem6  31852
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