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Theorem divdiv2 10257
Description: Division by a fraction. (Contributed by NM, 27-Dec-2008.)
Assertion
Ref Expression
divdiv2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  ( B  /  C ) )  =  ( ( A  x.  C )  /  B ) )

Proof of Theorem divdiv2
StepHypRef Expression
1 ax-1cn 9548 . . . . 5  |-  1  e.  CC
2 ax-1ne0 9559 . . . . 5  |-  1  =/=  0
31, 2pm3.2i 455 . . . 4  |-  ( 1  e.  CC  /\  1  =/=  0 )
4 divdivdiv 10246 . . . 4  |-  ( ( ( A  e.  CC  /\  ( 1  e.  CC  /\  1  =/=  0 ) )  /\  ( ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) ) )  ->  (
( A  /  1
)  /  ( B  /  C ) )  =  ( ( A  x.  C )  / 
( 1  x.  B
) ) )
53, 4mpanl2 681 . . 3  |-  ( ( A  e.  CC  /\  ( ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) ) )  ->  ( ( A  /  1 )  / 
( B  /  C
) )  =  ( ( A  x.  C
)  /  ( 1  x.  B ) ) )
653impb 1191 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  / 
1 )  /  ( B  /  C ) )  =  ( ( A  x.  C )  / 
( 1  x.  B
) ) )
7 div1 10237 . . . 4  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
873ad2ant1 1016 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  1
)  =  A )
98oveq1d 6292 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  / 
1 )  /  ( B  /  C ) )  =  ( A  / 
( B  /  C
) ) )
10 mulid2 9592 . . . . 5  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1110ad2antrl 727 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( 1  x.  B )  =  B )
12113adant3 1015 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  x.  B
)  =  B )
1312oveq2d 6293 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  C )  /  (
1  x.  B ) )  =  ( ( A  x.  C )  /  B ) )
146, 9, 133eqtr3d 2490 1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  ( B  /  C ) )  =  ( ( A  x.  C )  /  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636  (class class class)co 6277   CCcc 9488   0cc0 9490   1c1 9491    x. cmul 9495    / cdiv 10207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208
This theorem is referenced by:  divdiv2d  10353  aaliou3lem3  22605  chebbnd2  23527  dchrmusum2  23544  dchrvmasumlem2  23548  mulog2sumlem2  23585  pntibndlem3  23642  pntlemb  23647  pntlemn  23650  pntlemj  23653  pntlemf  23655  ofdivdiv2  31202  expgrowth  31209
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