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Theorem divdivdiv 10024
Description: Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.)
Assertion
Ref Expression
divdivdiv  |-  ( ( ( 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 divdivdiv
StepHypRef Expression
1 simprrl 763 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  D  e.  CC )
2 simprll 761 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  C  e.  CC )
3 simprlr 762 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  C  =/=  0 )
4 divcl 9992 . . . . . . 7  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( D  /  C )  e.  CC )
51, 2, 3, 4syl3anc 1218 . . . . . 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 753 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  A  e.  CC )
7 simplrl 759 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  B  e.  CC )
8 simplrr 760 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  B  =/=  0 )
9 divcl 9992 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
106, 7, 8, 9syl3anc 1218 . . . . . 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 9399 . . . . 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 754 . . . . . 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 755 . . . . . 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 divmuldiv 10023 . . . . . 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 1219 . . . . 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 2470 . . . 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 6102 . . 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 756 . . . . . . 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 divmuldiv 10023 . . . . . . 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 1219 . . . . . 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 9399 . . . . . . . 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 6101 . . . . . . 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 9398 . . . . . . . 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 764 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  D  =/=  0 )
251, 2, 24, 3mulne0d 9980 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( D  x.  C )  =/=  0 )
26 divid 10013 . . . . . . . 8  |-  ( ( ( D  x.  C
)  e.  CC  /\  ( D  x.  C
)  =/=  0 )  ->  ( ( D  x.  C )  / 
( D  x.  C
) )  =  1 )
2723, 25, 26syl2anc 661 . . . . . . 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 2470 . . . . . 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 2470 . . . . 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 6101 . . . 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 divcl 9992 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( C  /  D )  e.  CC )
322, 1, 24, 31syl3anc 1218 . . . . 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 9401 . . . 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 9396 . . . 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 2478 . . 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 2472 . 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 9398 . . . 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 9398 . . . 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 mulne0 9970 . . . . 5  |-  ( ( ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  x.  C
)  =/=  0 )
4039ad2ant2lr 747 . . . 4  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( B  x.  C )  =/=  0 )
41 divcl 9992 . . . 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 1218 . . 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 divne0 9998 . . . 4  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  /  D
)  =/=  0 )
4443adantl 466 . . 3  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( C  /  D )  =/=  0 )
45 divmul 9989 . . 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 1222 . 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 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275    x. cmul 9279    / cdiv 9985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986
This theorem is referenced by:  recdiv  10029  divcan7  10032  divdiv1  10034  divdiv2  10035  divdivdivi  10086  divdivdivd  10146  qreccl  10965  pnt2  22842
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