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

Theorem divdivdivap 8672
Description: Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
Assertion
Ref Expression
divdivdivap  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  B )  / 
( C  /  D
) )  =  ( ( A  x.  D
)  /  ( B  x.  C ) ) )

Proof of Theorem divdivdivap
StepHypRef Expression
1 simprrl 539 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  D  e.  CC )
2 simprll 537 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  C  e.  CC )
3 simprlr 538 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  C #  0 )
4 divclap 8637 . . . . . . 7  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( D  /  C )  e.  CC )
51, 2, 3, 4syl3anc 1238 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( D  /  C )  e.  CC )
6 simpll 527 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  A  e.  CC )
7 simplrl 535 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  B  e.  CC )
8 simplrr 536 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  B #  0 )
9 divclap 8637 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  e.  CC )
106, 7, 8, 9syl3anc 1238 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( A  /  B )  e.  CC )
115, 10mulcomd 7981 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( D  /  C )  x.  ( A  /  B
) )  =  ( ( A  /  B
)  x.  ( D  /  C ) ) )
12 simplr 528 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( B  e.  CC  /\  B #  0 ) )
13 simprl 529 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( C  e.  CC  /\  C #  0 ) )
14 divmuldivap 8671 . . . . . 6  |-  ( ( ( A  e.  CC  /\  D  e.  CC )  /\  ( ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) ) )  ->  ( ( A  /  B )  x.  ( D  /  C
) )  =  ( ( A  x.  D
)  /  ( B  x.  C ) ) )
156, 1, 12, 13, 14syl22anc 1239 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  B )  x.  ( D  /  C
) )  =  ( ( A  x.  D
)  /  ( B  x.  C ) ) )
1611, 15eqtrd 2210 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( D  /  C )  x.  ( A  /  B
) )  =  ( ( A  x.  D
)  /  ( B  x.  C ) ) )
1716oveq2d 5893 . . 3  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( C  /  D )  x.  ( ( D  /  C )  x.  ( A  /  B ) ) )  =  ( ( C  /  D )  x.  ( ( A  x.  D )  / 
( B  x.  C
) ) ) )
18 simprr 531 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( D  e.  CC  /\  D #  0 ) )
19 divmuldivap 8671 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  D  e.  CC )  /\  ( ( D  e.  CC  /\  D #  0 )  /\  ( C  e.  CC  /\  C #  0 ) ) )  ->  ( ( C  /  D )  x.  ( D  /  C
) )  =  ( ( C  x.  D
)  /  ( D  x.  C ) ) )
202, 1, 18, 13, 19syl22anc 1239 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( C  /  D )  x.  ( D  /  C
) )  =  ( ( C  x.  D
)  /  ( D  x.  C ) ) )
212, 1mulcomd 7981 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( C  x.  D )  =  ( D  x.  C ) )
2221oveq1d 5892 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( C  x.  D )  / 
( D  x.  C
) )  =  ( ( D  x.  C
)  /  ( D  x.  C ) ) )
231, 2mulcld 7980 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( D  x.  C )  e.  CC )
24 simprrr 540 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  D #  0 )
251, 2, 24, 3mulap0d 8617 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( D  x.  C ) #  0 )
26 dividap 8660 . . . . . . . 8  |-  ( ( ( D  x.  C
)  e.  CC  /\  ( D  x.  C
) #  0 )  -> 
( ( D  x.  C )  /  ( D  x.  C )
)  =  1 )
2723, 25, 26syl2anc 411 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( D  x.  C )  / 
( D  x.  C
) )  =  1 )
2822, 27eqtrd 2210 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( C  x.  D )  / 
( D  x.  C
) )  =  1 )
2920, 28eqtrd 2210 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( C  /  D )  x.  ( D  /  C
) )  =  1 )
3029oveq1d 5892 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( C  /  D )  x.  ( D  /  C ) )  x.  ( A  /  B
) )  =  ( 1  x.  ( A  /  B ) ) )
31 divclap 8637 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( C  /  D )  e.  CC )
322, 1, 24, 31syl3anc 1238 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( C  /  D )  e.  CC )
3332, 5, 10mulassd 7983 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( C  /  D )  x.  ( D  /  C ) )  x.  ( A  /  B
) )  =  ( ( C  /  D
)  x.  ( ( D  /  C )  x.  ( A  /  B ) ) ) )
3410mulid2d 7978 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( 1  x.  ( A  /  B
) )  =  ( A  /  B ) )
3530, 33, 343eqtr3d 2218 . . 3  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( C  /  D )  x.  ( ( D  /  C )  x.  ( A  /  B ) ) )  =  ( A  /  B ) )
3617, 35eqtr3d 2212 . 2  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( C  /  D )  x.  ( ( A  x.  D )  /  ( B  x.  C )
) )  =  ( A  /  B ) )
376, 1mulcld 7980 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( A  x.  D )  e.  CC )
387, 2mulcld 7980 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( B  x.  C )  e.  CC )
39 mulap0 8613 . . . . 5  |-  ( ( ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( B  x.  C ) #  0 )
4039ad2ant2lr 510 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( B  x.  C ) #  0 )
41 divclap 8637 . . . 4  |-  ( ( ( A  x.  D
)  e.  CC  /\  ( B  x.  C
)  e.  CC  /\  ( B  x.  C
) #  0 )  -> 
( ( A  x.  D )  /  ( B  x.  C )
)  e.  CC )
4237, 38, 40, 41syl3anc 1238 . . 3  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  x.  D )  / 
( B  x.  C
) )  e.  CC )
43 divap0 8643 . . . 4  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  /  D ) #  0 )
4443adantl 277 . . 3  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( C  /  D ) #  0 )
45 divmulap 8634 . . 3  |-  ( ( ( A  /  B
)  e.  CC  /\  ( ( A  x.  D )  /  ( B  x.  C )
)  e.  CC  /\  ( ( C  /  D )  e.  CC  /\  ( C  /  D
) #  0 ) )  ->  ( ( ( A  /  B )  /  ( C  /  D ) )  =  ( ( A  x.  D )  /  ( B  x.  C )
)  <->  ( ( C  /  D )  x.  ( ( A  x.  D )  /  ( B  x.  C )
) )  =  ( A  /  B ) ) )
4610, 42, 32, 44, 45syl112anc 1242 . 2  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( A  /  B )  /  ( C  /  D ) )  =  ( ( A  x.  D )  /  ( B  x.  C )
)  <->  ( ( C  /  D )  x.  ( ( A  x.  D )  /  ( B  x.  C )
) )  =  ( A  /  B ) ) )
4736, 46mpbird 167 1  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  B )  / 
( C  /  D
) )  =  ( ( A  x.  D
)  /  ( B  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814    x. cmul 7818   # cap 8540    / cdiv 8631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632
This theorem is referenced by:  recdivap  8677  divcanap7  8680  divdivap1  8682  divdivap2  8683  divdivdivapi  8734  qreccl  9644  pcadd  12341
  Copyright terms: Public domain W3C validator