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Theorem divcan5 9717
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 9706 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( C  /  C
)  =  1 )
21oveq1d 6097 . . 3  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( ( C  /  C )  x.  ( A  /  B ) )  =  ( 1  x.  ( A  /  B
) ) )
323ad2ant3 981 . 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 986 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
5 simp1 958 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  A  e.  CC )
6 simp3 960 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
7 simp2 959 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
8 divmuldiv 9715 . . 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 1186 . 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 9685 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
11103expb 1155 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  CC )
1211mulid2d 9107 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( 1  x.  ( A  /  B ) )  =  ( A  /  B
) )
13123adant3 978 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  x.  ( A  /  B ) )  =  ( A  /  B ) )
143, 9, 133eqtr3d 2477 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600  (class class class)co 6082   CCcc 8989   0cc0 8991   1c1 8992    x. cmul 8996    / cdiv 9678
This theorem is referenced by:  divcan7  9724  divadddiv  9730  divcan5d  9817  8th4div3  10192  modmulnn  11266  moddi  11285  reccn2  12391  efif1olem4  20448  ang180lem1  20652  quart1  20697  divsqrsumlem  20819  basellem1  20864  ppiub  20989  bposlem8  21076  chpchtlim  21174  pnt2  21308  bpoly3  26105  heiborlem6  26526
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679
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