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Theorem divcan5 10247
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 10235 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( C  /  C
)  =  1 )
21oveq1d 6300 . . 3  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( ( C  /  C )  x.  ( A  /  B ) )  =  ( 1  x.  ( A  /  B
) ) )
323ad2ant3 1019 . 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 1024 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
5 simp1 996 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  A  e.  CC )
6 simp3 998 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
7 simp2 997 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
8 divmuldiv 10245 . . 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 1229 . 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 10214 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
11103expb 1197 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  CC )
1211mulid2d 9615 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( 1  x.  ( A  /  B ) )  =  ( A  /  B
) )
13123adant3 1016 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  x.  ( A  /  B ) )  =  ( A  /  B ) )
143, 9, 133eqtr3d 2516 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6285   CCcc 9491   0cc0 9493   1c1 9494    x. cmul 9498    / cdiv 10207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208
This theorem is referenced by:  divcan7  10254  divadddiv  10260  divcan5d  10347  8th4div3  10760  modmulnn  11982  moddi  12023  reccn2  13385  efif1olem4  22757  ang180lem1  22966  quart1  23012  divsqrtsumlem  23134  basellem1  23179  ppiub  23304  bposlem8  23391  chpchtlim  23489  pnt2  23623  bpoly3  29673  dvasin  29956  heiborlem6  30142
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