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

Theorem divdivdiv 10241
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 10209 . . . . . . 7  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( D  /  C )  e.  CC )
51, 2, 3, 4syl3anc 1228 . . . . . 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 10209 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
106, 7, 8, 9syl3anc 1228 . . . . . 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 9613 . . . . 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 10240 . . . . . 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 1229 . . . . 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 2508 . . . 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 6298 . . 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 10240 . . . . . . 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 1229 . . . . . 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 9613 . . . . . . . 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 6297 . . . . . . 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 9612 . . . . . . . 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 10197 . . . . . . . 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 10230 . . . . . . . 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 2508 . . . . . 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 2508 . . . . 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 6297 . . . 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 10209 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( C  /  D )  e.  CC )
322, 1, 24, 31syl3anc 1228 . . . . 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 9615 . . . 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 9610 . . . 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 2516 . . 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 2510 . 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 9612 . . . 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 9612 . . . 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 10187 . . . . 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 10209 . . . 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 1228 . . 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 10215 . . . 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 10206 . . 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 1232 . 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 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    x. cmul 9493    / cdiv 10202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203
This theorem is referenced by:  recdiv  10246  divcan7  10249  divdiv1  10251  divdiv2  10252  divdivdivi  10303  divdivdivd  10363  qreccl  11198  pnt2  23523
  Copyright terms: Public domain W3C validator