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Theorem divmuldiv 10306
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 986 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) ) )
2 3anass 986 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )
3 divcl 10275 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
4 divcl 10275 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 9622 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 479 . . . . 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 9622 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 751 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
983adantr1 1164 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1161 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulne0 10253 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
12113adantr1 1164 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
13123adantl1 1161 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( C  x.  D )  =/=  0
)
14 divcan3 10293 . . . . 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 1264 . . . 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 1006 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  C  e.  CC )
1716, 3jca 534 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 1006 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  D  e.  CC )
1918, 4jca 534 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 9801 . . . . . . 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 479 . . . . . 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 10277 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcan2 10277 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 6324 . . . . . 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 2472 . . . . 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 6320 . . . 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 2472 . . 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 482 . 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 833 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625  (class class class)co 6305   CCcc 9536   0cc0 9538    x. cmul 9543    / cdiv 10268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269
This theorem is referenced by:  divdivdiv  10307  divcan5  10308  divmul13  10309  divmul24  10310  divmuldivi  10366  divmuldivd  10423  qmulcl  11282  mulexpz  12309  expaddz  12313  sqdiv  12337  faclbnd2  12473  bcm1k  12497  bcp1n  12498  pythagtriplem16  14743  dvsqrt  23555  dquartlem1  23650  basellem8  23885  dchrvmasumlem1  24204  dchrvmasum2lem  24205  pntlemr  24311  pntlemf  24314  wallispilem4  37514
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