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Theorem divdiv32 10027
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 9989 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
2 div23 10001 . . 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 1245 . 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 9998 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
543expb 1181 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
653adant3 1001 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
76oveq1d 6095 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  B )  /  C
)  =  ( ( A  x.  ( 1  /  B ) )  /  C ) )
8 divcl 9988 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
983expb 1181 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
10 divrec 9998 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  C
)  /  B )  =  ( ( A  /  C )  x.  ( 1  /  B
) ) )
119, 10syl3an1 1244 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  C
)  /  B )  =  ( ( A  /  C )  x.  ( 1  /  B
) ) )
12113expb 1181 . . . 4  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  C )  /  B
)  =  ( ( A  /  C )  x.  ( 1  /  B ) ) )
13123impa 1175 . . 3  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  C )  /  B
)  =  ( ( A  /  C )  x.  ( 1  /  B ) ) )
14133com23 1186 . 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 2475 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 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:  divdiv23zi  10072  divdiv32d  10120  efeq1  21870  logexprlim  22449  pntlemb  22731
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