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Theorem divdiv32 10264
Description: Swap denominators in a division. (Contributed by NM, 2-Aug-2004.)
Assertion
Ref Expression
divdiv32  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  C
)  =  ( ( A  /  C )  /  B ) )

Proof of Theorem divdiv32
StepHypRef Expression
1 reccl 10226 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
2 div23 10238 . . 3  |-  ( ( A  e.  CC  /\  ( 1  /  B
)  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  x.  ( 1  /  B ) )  /  C )  =  ( ( A  /  C )  x.  (
1  /  B ) ) )
31, 2syl3an2 1262 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  ( 1  /  B
) )  /  C
)  =  ( ( A  /  C )  x.  ( 1  /  B ) ) )
4 divrec 10235 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
543expb 1197 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
653adant3 1016 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
76oveq1d 6310 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  C
)  =  ( ( A  x.  ( 1  /  B ) )  /  C ) )
8 divcl 10225 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
983expb 1197 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
10 divrec 10235 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  C
)  /  B )  =  ( ( A  /  C )  x.  ( 1  /  B
) ) )
119, 10syl3an1 1261 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  C
)  /  B )  =  ( ( A  /  C )  x.  ( 1  /  B
) ) )
12113expb 1197 . . . 4  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  C )  /  B
)  =  ( ( A  /  C )  x.  ( 1  /  B ) ) )
13123impa 1191 . . 3  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  C )  /  B
)  =  ( ( A  /  C )  x.  ( 1  /  B ) ) )
14133com23 1202 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  /  B
)  =  ( ( A  /  C )  x.  ( 1  /  B ) ) )
153, 7, 143eqtr4d 2518 1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  C
)  =  ( ( A  /  C )  /  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 6295   CCcc 9502   0cc0 9504   1c1 9505    x. cmul 9509    / cdiv 10218
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219
This theorem is referenced by:  divdiv23zi  10309  divdiv32d  10357  efeq1  22782  logexprlim  23366  pntlemb  23648
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