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Theorem divmuldiv 9706
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 9676 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
4 divcl 9676 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 9066 . . . . . 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 9066 . . . . . . . 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 9656 . . . . . . 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 9694 . . . . 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 9227 . . . . . . 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 9678 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcan2 9678 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 6092 . . . . . 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 2469 . . . . 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 6088 . . . 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 2469 . . 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 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   0cc0 8982    x. cmul 8987    / cdiv 9669
This theorem is referenced by:  divdivdiv  9707  divcan5  9708  divmul13  9709  divmul24  9710  divmuldivi  9766  divmuldivd  9823  qmulcl  10584  mulexpz  11412  expaddz  11416  sqdiv  11439  faclbnd2  11574  bcm1k  11598  bcp1n  11599  pythagtriplem16  13196  dvsqr  20620  dquartlem1  20683  basellem8  20862  dchrvmasumlem1  21181  dchrvmasum2lem  21182  pntlemr  21288  pntlemf  21291  wallispilem4  27784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670
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