MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divdivdiv Unicode version

Theorem divdivdiv 9608
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 9577 . . . . . . 7  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( D  /  C )  e.  CC )
51, 2, 3, 4syl3anc 1183 . . . . . 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 9577 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
106, 7, 8, 9syl3anc 1183 . . . . . 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 9003 . . . . 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 9607 . . . . . 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 1184 . . . . 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 2398 . . . 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 5997 . . 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 9607 . . . . . . 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 1184 . . . . . 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 9003 . . . . . . . 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 5996 . . . . . . 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 9002 . . . . . . . 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 9567 . . . . . . . 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 9598 . . . . . . . 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 2398 . . . . . 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 2398 . . . . 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 5996 . . . 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 9577 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( C  /  D )  e.  CC )
322, 1, 24, 31syl3anc 1183 . . . . 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 9005 . . . 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 9000 . . . 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 2406 . . 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 2400 . 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 9002 . . . 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 9002 . . . 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 9557 . . . . 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 9577 . . . 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 1183 . . 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 9583 . . . 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 9574 . . 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 1187 . 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 1647    e. wcel 1715    =/= wne 2529  (class class class)co 5981   CCcc 8882   0cc0 8884   1c1 8885    x. cmul 8889    / cdiv 9570
This theorem is referenced by:  recdiv  9613  divcan7  9616  divdiv1  9618  divdiv2  9619  divdivdivi  9670  divdivdivd  9730  qreccl  10487  pnt2  20985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571
  Copyright terms: Public domain W3C validator