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Theorem divmuldiv 9670
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 940 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) ) )
2 3anass 940 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )
3 divcl 9640 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
4 divcl 9640 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 9030 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 464 . . . . 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 9030 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 728 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
983adantr1 1116 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1113 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulne0 9620 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
12113adantr1 1116 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
13123adantl1 1113 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( C  x.  D )  =/=  0
)
14 divcan3 9658 . . . . 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 1184 . . . 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 958 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  C  e.  CC )
1716, 3jca 519 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 958 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  D  e.  CC )
1918, 4jca 519 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 9191 . . . . . . 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 464 . . . . . 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 9642 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcan2 9642 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 6059 . . . . . 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 2438 . . . . 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 6055 . . . 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 2438 . . 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 467 . 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 800 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567  (class class class)co 6040   CCcc 8944   0cc0 8946    x. cmul 8951    / cdiv 9633
This theorem is referenced by:  divdivdiv  9671  divcan5  9672  divmul13  9673  divmul24  9674  divmuldivi  9730  divmuldivd  9787  qmulcl  10548  mulexpz  11375  expaddz  11379  sqdiv  11402  faclbnd2  11537  bcm1k  11561  bcp1n  11562  pythagtriplem16  13159  dvsqr  20581  dquartlem1  20644  basellem8  20823  dchrvmasumlem1  21142  dchrvmasum2lem  21143  pntlemr  21249  pntlemf  21252  wallispilem4  27684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634
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