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Theorem divmuldiv 10145
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 969 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) ) )
2 3anass 969 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )
3 divcl 10114 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
4 divcl 10114 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 9480 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 477 . . . . 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 9480 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 746 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
983adantr1 1147 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1144 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulne0 10092 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
12113adantr1 1147 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
13123adantl1 1144 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 ) )  ->  ( C  x.  D )  =/=  0
)
14 divcan3 10132 . . . . 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 1219 . . . 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 989 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  C  e.  CC )
1716, 3jca 532 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 989 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  D  e.  CC )
1918, 4jca 532 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 9652 . . . . . . 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 477 . . . . . 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 10116 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcan2 10116 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D  =/=  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 6222 . . . . . 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 2497 . . . . 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 6218 . . . 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 2497 . . 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 480 . 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 822 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648  (class class class)co 6203   CCcc 9394   0cc0 9396    x. cmul 9401    / cdiv 10107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108
This theorem is referenced by:  divdivdiv  10146  divcan5  10147  divmul13  10148  divmul24  10149  divmuldivi  10205  divmuldivd  10262  qmulcl  11085  mulexpz  12024  expaddz  12028  sqdiv  12051  faclbnd2  12187  bcm1k  12211  bcp1n  12212  pythagtriplem16  14018  dvsqr  22318  dquartlem1  22382  basellem8  22561  dchrvmasumlem1  22880  dchrvmasum2lem  22881  pntlemr  22987  pntlemf  22990  wallispilem4  30031
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