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Theorem ddcan 9661
Description: Cancellation in a double division. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
ddcan  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  /  ( A  /  B ) )  =  B )

Proof of Theorem ddcan
StepHypRef Expression
1 simpll 731 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  A  e.  CC )
2 simprl 733 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  B  e.  CC )
3 simprr 734 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  B  =/=  0 )
4 divcan1 9620 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  x.  B )  =  A )
51, 2, 3, 4syl3anc 1184 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  B )  x.  B
)  =  A )
6 divcl 9617 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
71, 2, 3, 6syl3anc 1184 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  /  B
)  e.  CC )
8 divne0 9623 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  /  B
)  =/=  0 )
9 divmul 9614 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  (
( A  /  B
)  e.  CC  /\  ( A  /  B
)  =/=  0 ) )  ->  ( ( A  /  ( A  /  B ) )  =  B  <->  ( ( A  /  B )  x.  B )  =  A ) )
101, 2, 7, 8, 9syl112anc 1188 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  / 
( A  /  B
) )  =  B  <-> 
( ( A  /  B )  x.  B
)  =  A ) )
115, 10mpbird 224 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  /  ( A  /  B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551  (class class class)co 6021   CCcc 8922   0cc0 8924    x. cmul 8929    / cdiv 9610
This theorem is referenced by:  ddcand  9743  fsumdvdscom  20838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611
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