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

Theorem divdivdivap 8234
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 507 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  D  e.  CC )
2 simprll 505 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  C  e.  CC )
3 simprlr 506 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  C #  0 )
4 divclap 8199 . . . . . . 7  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( D  /  C )  e.  CC )
51, 2, 3, 4syl3anc 1175 . . . . . 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 497 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  A  e.  CC )
7 simplrl 503 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  B  e.  CC )
8 simplrr 504 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  B #  0 )
9 divclap 8199 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  e.  CC )
106, 7, 8, 9syl3anc 1175 . . . . . 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 7563 . . . . 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 498 . . . . . 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 499 . . . . . 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 8233 . . . . . 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 1176 . . . . 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 2121 . . . 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 5682 . . 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 500 . . . . . . 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 8233 . . . . . . 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 1176 . . . . . 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 7563 . . . . . . . 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 5681 . . . . . . 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 7562 . . . . . . . 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 508 . . . . . . . . 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 8181 . . . . . . . 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 8222 . . . . . . . 8  |-  ( ( ( D  x.  C
)  e.  CC  /\  ( D  x.  C
) #  0 )  -> 
( ( D  x.  C )  /  ( D  x.  C )
)  =  1 )
2723, 25, 26syl2anc 404 . . . . . . 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 2121 . . . . . 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 2121 . . . . 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 5681 . . . 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 8199 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( C  /  D )  e.  CC )
322, 1, 24, 31syl3anc 1175 . . . . 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 7565 . . . 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 7560 . . . 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 2129 . . 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 2123 . 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 7562 . . . 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 7562 . . . 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 8177 . . . . 5  |-  ( ( ( B  e.  CC  /\  B #  0 )  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( B  x.  C ) #  0 )
4039ad2ant2lr 495 . . . 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 8199 . . . 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 1175 . . 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 8205 . . . 4  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  /  D ) #  0 )
4443adantl 272 . . 3  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( C  /  D ) #  0 )
45 divmulap 8196 . . 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 1179 . 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 166 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 103    <-> wb 104    = wceq 1290    e. wcel 1439   class class class wbr 3851  (class class class)co 5666   CCcc 7402   0cc0 7404   1c1 7405    x. cmul 7409   # cap 8112    / cdiv 8193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194
This theorem is referenced by:  recdivap  8239  divcanap7  8242  divdivap1  8244  divdivap2  8245  divdivdivapi  8296  qreccl  9181
  Copyright terms: Public domain W3C validator