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Theorem divmuldiv 9462
Description: Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.)
Assertion
Ref Expression
divmuldiv  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  =  ( ( A  x.  B
)  /  ( C  x.  D ) ) )

Proof of Theorem divmuldiv
StepHypRef Expression
1 3anass 938 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) ) )
2 3anass 938 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )
3 divcl 9432 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
4 divcl 9432 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 8823 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 463 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
7 mulcl 8823 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 727 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
983adantr1 1114 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1111 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulne0 9412 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
12113adantr1 1114 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
13123adantl1 1111 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( C  x.  D )  =/=  0
)
14 divcan3 9450 . . . . 5  |-  ( ( ( ( A  /  C )  x.  ( B  /  D ) )  e.  CC  /\  ( C  x.  D )  e.  CC  /\  ( C  x.  D )  =/=  0 )  ->  (
( ( C  x.  D )  x.  (
( A  /  C
)  x.  ( B  /  D ) ) )  /  ( C  x.  D ) )  =  ( ( A  /  C )  x.  ( B  /  D
) ) )
156, 10, 13, 14syl3anc 1182 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( ( ( C  x.  D )  x.  ( ( A  /  C )  x.  ( B  /  D
) ) )  / 
( C  x.  D
) )  =  ( ( A  /  C
)  x.  ( B  /  D ) ) )
16 simp2 956 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  C  e.  CC )
1716, 3jca 518 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 956 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  D  e.  CC )
1918, 4jca 518 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 8983 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  ( A  /  C
)  e.  CC )  /\  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )  -> 
( ( C  x.  ( A  /  C
) )  x.  ( D  x.  ( B  /  D ) ) )  =  ( ( C  x.  D )  x.  ( ( A  /  C )  x.  ( B  /  D ) ) ) )
2117, 19, 20syl2an 463 . . . . . 6  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( ( C  x.  ( A  /  C ) )  x.  ( D  x.  ( B  /  D ) ) )  =  ( ( C  x.  D )  x.  ( ( A  /  C )  x.  ( B  /  D
) ) ) )
22 divcan2 9434 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcan2 9434 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 5879 . . . . . 6  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( ( C  x.  ( A  /  C ) )  x.  ( D  x.  ( B  /  D ) ) )  =  ( A  x.  B ) )
2521, 24eqtr3d 2319 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( ( C  x.  D )  x.  ( ( A  /  C )  x.  ( B  /  D ) ) )  =  ( A  x.  B ) )
2625oveq1d 5875 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( ( ( C  x.  D )  x.  ( ( A  /  C )  x.  ( B  /  D
) ) )  / 
( C  x.  D
) )  =  ( ( A  x.  B
)  /  ( C  x.  D ) ) )
2715, 26eqtr3d 2319 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  =  ( ( A  x.  B
)  /  ( C  x.  D ) ) )
281, 2, 27syl2anbr 466 . 2  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  /\  ( B  e.  CC  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  =  ( ( A  x.  B
)  /  ( C  x.  D ) ) )
2928an4s 799 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  =  ( ( A  x.  B
)  /  ( C  x.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448  (class class class)co 5860   CCcc 8737   0cc0 8739    x. cmul 8744    / cdiv 9425
This theorem is referenced by:  divdivdiv  9463  divcan5  9464  divmul13  9465  divmul24  9466  divmuldivi  9522  divmuldivd  9579  qmulcl  10336  mulexpz  11144  expaddz  11148  sqdiv  11171  faclbnd2  11306  bcm1k  11329  bcp1n  11330  pythagtriplem16  12885  dvsqr  20086  dquartlem1  20149  basellem8  20327  dchrvmasumlem1  20646  dchrvmasum2lem  20647  pntlemr  20753  pntlemf  20756  wallispilem4  27828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426
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