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Theorem divcan5 10045
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 10033 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( C  /  C
)  =  1 )
21oveq1d 6118 . . 3  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( ( C  /  C )  x.  ( A  /  B ) )  =  ( 1  x.  ( A  /  B
) ) )
323ad2ant3 1011 . 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 1016 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
5 simp1 988 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  A  e.  CC )
6 simp3 990 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
7 simp2 989 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
8 divmuldiv 10043 . . 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 1219 . 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 10012 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
11103expb 1188 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  CC )
1211mulid2d 9416 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( 1  x.  ( A  /  B ) )  =  ( A  /  B
) )
13123adant3 1008 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  x.  ( A  /  B ) )  =  ( A  /  B ) )
143, 9, 133eqtr3d 2483 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618  (class class class)co 6103   CCcc 9292   0cc0 9294   1c1 9295    x. cmul 9299    / cdiv 10005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006
This theorem is referenced by:  divcan7  10052  divadddiv  10058  divcan5d  10145  8th4div3  10557  modmulnn  11737  moddi  11778  reccn2  13086  efif1olem4  22013  ang180lem1  22217  quart1  22263  divsqrsumlem  22385  basellem1  22430  ppiub  22555  bposlem8  22642  chpchtlim  22740  pnt2  22874  bpoly3  28213  dvasin  28492  heiborlem6  28727
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