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Theorem divdiv2 10046
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 9343 . . . . 5  |-  1  e.  CC
2 ax-1ne0 9354 . . . . 5  |-  1  =/=  0
31, 2pm3.2i 455 . . . 4  |-  ( 1  e.  CC  /\  1  =/=  0 )
4 divdivdiv 10035 . . . 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 1183 . 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 10026 . . . 4  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
873ad2ant1 1009 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  1
)  =  A )
98oveq1d 6109 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  / 
1 )  /  ( B  /  C ) )  =  ( A  / 
( B  /  C
) ) )
10 mulid2 9387 . . . . 5  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1110ad2antrl 727 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( 1  x.  B )  =  B )
12113adant3 1008 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  x.  B
)  =  B )
1312oveq2d 6110 . 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 2483 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 965    = wceq 1369    e. wcel 1756    =/= wne 2609  (class class class)co 6094   CCcc 9283   0cc0 9285   1c1 9286    x. cmul 9290    / cdiv 9996
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 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
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 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-po 4644  df-so 4645  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997
This theorem is referenced by:  divdiv2d  10142  aaliou3lem3  21813  chebbnd2  22729  dchrmusum2  22746  dchrvmasumlem2  22750  mulog2sumlem2  22787  pntibndlem3  22844  pntlemb  22849  pntlemn  22852  pntlemj  22855  pntlemf  22857  ofdivdiv2  29605  expgrowth  29612
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