MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divdiv2 Structured version   Unicode version

Theorem divdiv2 10031
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 9328 . . . . 5  |-  1  e.  CC
2 ax-1ne0 9339 . . . . 5  |-  1  =/=  0
31, 2pm3.2i 452 . . . 4  |-  ( 1  e.  CC  /\  1  =/=  0 )
4 divdivdiv 10020 . . . 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 674 . . 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 1176 . 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 10011 . . . 4  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
873ad2ant1 1002 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  1
)  =  A )
98oveq1d 6095 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  / 
1 )  /  ( B  /  C ) )  =  ( A  / 
( B  /  C
) ) )
10 mulid2 9372 . . . . 5  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1110ad2antrl 720 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( 1  x.  B )  =  B )
12113adant3 1001 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  x.  B
)  =  B )
1312oveq2d 6096 . 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 2473 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596  (class class class)co 6080   CCcc 9268   0cc0 9270   1c1 9271    x. cmul 9275    / cdiv 9981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982
This theorem is referenced by:  divdiv2d  10127  aaliou3lem3  21695  chebbnd2  22611  dchrmusum2  22628  dchrvmasumlem2  22632  mulog2sumlem2  22669  pntibndlem3  22726  pntlemb  22731  pntlemn  22734  pntlemj  22737  pntlemf  22739  ofdivdiv2  29447  expgrowth  29454
  Copyright terms: Public domain W3C validator