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Theorem divdivap2 8189
Description: Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
Assertion
Ref Expression
divdivap2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  / 
( B  /  C
) )  =  ( ( A  x.  C
)  /  B ) )

Proof of Theorem divdivap2
StepHypRef Expression
1 ax-1cn 7436 . . . . 5  |-  1  e.  CC
2 1ap0 8065 . . . . 5  |-  1 #  0
31, 2pm3.2i 266 . . . 4  |-  ( 1  e.  CC  /\  1 #  0 )
4 divdivdivap 8178 . . . 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 426 . . 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 1139 . 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 8168 . . . 4  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
873ad2ant1 964 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  / 
1 )  =  A )
98oveq1d 5667 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  1 )  / 
( B  /  C
) )  =  ( A  /  ( B  /  C ) ) )
10 mulid2 7484 . . . . 5  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1110ad2antrl 474 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( 1  x.  B )  =  B )
12113adant3 963 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( 1  x.  B )  =  B )
1312oveq2d 5668 . 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 2128 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 set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7346   0cc0 7348   1c1 7349    x. cmul 7353   # cap 8056    / cdiv 8137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-po 4123  df-iso 4124  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138
This theorem is referenced by:  divdivap2d  8288
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