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Theorem divdivdiv 9548
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 740 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  D  e.  CC )
2 simprll 738 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  C  e.  CC )
3 simprlr 739 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  C  =/=  0 )
4 divcl 9517 . . . . . . 7  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( D  /  C )  e.  CC )
51, 2, 3, 4syl3anc 1182 . . . . . 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 730 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  A  e.  CC )
7 simplrl 736 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  B  e.  CC )
8 simplrr 737 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  B  =/=  0 )
9 divcl 9517 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
106, 7, 8, 9syl3anc 1182 . . . . . 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 8943 . . . . 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 731 . . . . . 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 732 . . . . . 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 9547 . . . . . 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 1183 . . . . 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 2390 . . . 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 5958 . . 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 733 . . . . . . 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 9547 . . . . . . 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 1183 . . . . . 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 8943 . . . . . . . 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 5957 . . . . . . 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 8942 . . . . . . . 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 741 . . . . . . . . 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 9507 . . . . . . . 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 9538 . . . . . . . 8  |-  ( ( ( D  x.  C
)  e.  CC  /\  ( D  x.  C
)  =/=  0 )  ->  ( ( D  x.  C )  / 
( D  x.  C
) )  =  1 )
2723, 25, 26syl2anc 642 . . . . . . 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 2390 . . . . . 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 2390 . . . . 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 5957 . . . 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 9517 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( C  /  D )  e.  CC )
322, 1, 24, 31syl3anc 1182 . . . . 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 8945 . . . 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 8940 . . . 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 2398 . . 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 2392 . 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 8942 . . . 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 8942 . . . 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 9497 . . . . 5  |-  ( ( ( B  e.  CC  /\  B  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  x.  C
)  =/=  0 )
4039ad2ant2lr 728 . . . 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 9517 . . . 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 1182 . . 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 9523 . . . 4  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  /  D
)  =/=  0 )
4443adantl 452 . . 3  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( C  /  D )  =/=  0 )
45 divmul 9514 . . 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 1186 . 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 223 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    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521  (class class class)co 5942   CCcc 8822   0cc0 8824   1c1 8825    x. cmul 8829    / cdiv 9510
This theorem is referenced by:  recdiv  9553  divcan7  9556  divdiv1  9558  divdiv2  9559  divdivdivi  9610  divdivdivd  9670  qreccl  10425  pnt2  20868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511
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