ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  divdiv32ap Unicode version

Theorem divdiv32ap 8503
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 8462 . . 3  |-  ( ( B  e.  CC  /\  B #  0 )  ->  (
1  /  B )  e.  CC )
2 div23ap 8474 . . 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 1251 . 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 8471 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
543expb 1183 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
653adant3 1002 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
76oveq1d 5796 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  /  C )  =  ( ( A  x.  (
1  /  B ) )  /  C ) )
8 divclap 8461 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
983expb 1183 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  C )  e.  CC )
10 divrecap 8471 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  (
( A  /  C
)  /  B )  =  ( ( A  /  C )  x.  ( 1  /  B
) ) )
119, 10syl3an1 1250 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  /\  B  e.  CC  /\  B #  0 )  -> 
( ( A  /  C )  /  B
)  =  ( ( A  /  C )  x.  ( 1  /  B ) ) )
12113expb 1183 . . . 4  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  (
( A  /  C
)  /  B )  =  ( ( A  /  C )  x.  ( 1  /  B
) ) )
13123impa 1177 . . 3  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( A  /  C )  /  B )  =  ( ( A  /  C
)  x.  ( 1  /  B ) ) )
14133com23 1188 . 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 2183 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 103    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3936  (class class class)co 5781   CCcc 7641   0cc0 7643   1c1 7644    x. cmul 7648   # cap 8366    / cdiv 8455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-po 4225  df-iso 4226  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456
This theorem is referenced by:  divdiv23apzi  8548  divdiv32apd  8599
  Copyright terms: Public domain W3C validator