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Theorem divdiv32ap 8103
Description: Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
Assertion
Ref Expression
divdiv32ap  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  /  C )  =  ( ( A  /  C
)  /  B ) )

Proof of Theorem divdiv32ap
StepHypRef Expression
1 recclap 8062 . . 3  |-  ( ( B  e.  CC  /\  B #  0 )  ->  (
1  /  B )  e.  CC )
2 div23ap 8074 . . 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 1206 . 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 divrecap 8071 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
543expb 1142 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
653adant3 961 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
76oveq1d 5609 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  /  C )  =  ( ( A  x.  (
1  /  B ) )  /  C ) )
8 divclap 8061 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
983expb 1142 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  C )  e.  CC )
10 divrecap 8071 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  (
( A  /  C
)  /  B )  =  ( ( A  /  C )  x.  ( 1  /  B
) ) )
119, 10syl3an1 1205 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  /\  B  e.  CC  /\  B #  0 )  -> 
( ( A  /  C )  /  B
)  =  ( ( A  /  C )  x.  ( 1  /  B ) ) )
12113expb 1142 . . . 4  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  (
( A  /  C
)  /  B )  =  ( ( A  /  C )  x.  ( 1  /  B
) ) )
13123impa 1136 . . 3  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( A  /  C )  /  B )  =  ( ( A  /  C
)  x.  ( 1  /  B ) ) )
14133com23 1147 . 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 2127 1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  /  C )  =  ( ( A  /  C
)  /  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 922    = wceq 1287    e. wcel 1436   class class class wbr 3814  (class class class)co 5594   CCcc 7269   0cc0 7271   1c1 7272    x. cmul 7276   # cap 7976    / cdiv 8055
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-id 4087  df-po 4090  df-iso 4091  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-iota 4937  df-fun 4974  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056
This theorem is referenced by:  divdiv23apzi  8148  divdiv32apd  8197
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