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Theorem divadddiv 9685
Description: Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 2-May-2016.)
Assertion
Ref Expression
divadddiv  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( A  /  C )  +  ( B  /  D
) )  =  ( ( ( A  x.  D )  +  ( B  x.  C ) )  /  ( C  x.  D ) ) )

Proof of Theorem divadddiv
StepHypRef Expression
1 mulcl 9030 . . . . 5  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  D
)  e.  CC )
21ad2ant2r 728 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( A  x.  D
)  e.  CC )
32adantrl 697 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( A  x.  D )  e.  CC )
4 mulcl 9030 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  e.  CC )
54adantrr 698 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( B  x.  C )  e.  CC )
65ad2ant2lr 729 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( B  x.  C )  e.  CC )
7 mulcl 9030 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 728 . . . . 5  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  e.  CC )
9 mulne0 9620 . . . . 5  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( C  x.  D
)  =/=  0 )
108, 9jca 519 . . . 4  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( ( C  x.  D )  e.  CC  /\  ( C  x.  D
)  =/=  0 ) )
1110adantl 453 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( C  x.  D )  e.  CC  /\  ( C  x.  D )  =/=  0 ) )
12 divdir 9657 . . 3  |-  ( ( ( A  x.  D
)  e.  CC  /\  ( B  x.  C
)  e.  CC  /\  ( ( C  x.  D )  e.  CC  /\  ( C  x.  D
)  =/=  0 ) )  ->  ( (
( A  x.  D
)  +  ( B  x.  C ) )  /  ( C  x.  D ) )  =  ( ( ( A  x.  D )  / 
( C  x.  D
) )  +  ( ( B  x.  C
)  /  ( C  x.  D ) ) ) )
133, 6, 11, 12syl3anc 1184 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( ( A  x.  D )  +  ( B  x.  C ) )  / 
( C  x.  D
) )  =  ( ( ( A  x.  D )  /  ( C  x.  D )
)  +  ( ( B  x.  C )  /  ( C  x.  D ) ) ) )
14 simpll 731 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  A  e.  CC )
15 simprr 734 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( D  e.  CC  /\  D  =/=  0 ) )
1615simpld 446 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  D  e.  CC )
1714, 16mulcomd 9065 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( A  x.  D )  =  ( D  x.  A ) )
18 simprll 739 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  C  e.  CC )
1918, 16mulcomd 9065 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( C  x.  D )  =  ( D  x.  C ) )
2017, 19oveq12d 6058 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( A  x.  D )  / 
( C  x.  D
) )  =  ( ( D  x.  A
)  /  ( D  x.  C ) ) )
21 simprl 733 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( C  e.  CC  /\  C  =/=  0 ) )
22 divcan5 9672 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( ( D  x.  A )  /  ( D  x.  C )
)  =  ( A  /  C ) )
2314, 21, 15, 22syl3anc 1184 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( D  x.  A )  / 
( D  x.  C
) )  =  ( A  /  C ) )
2420, 23eqtrd 2436 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( A  x.  D )  / 
( C  x.  D
) )  =  ( A  /  C ) )
25 simplr 732 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  B  e.  CC )
2625, 18mulcomd 9065 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( B  x.  C )  =  ( C  x.  B ) )
2726oveq1d 6055 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( B  x.  C )  / 
( C  x.  D
) )  =  ( ( C  x.  B
)  /  ( C  x.  D ) ) )
28 divcan5 9672 . . . . 5  |-  ( ( B  e.  CC  /\  ( D  e.  CC  /\  D  =/=  0 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  x.  B )  /  ( C  x.  D )
)  =  ( B  /  D ) )
2925, 15, 21, 28syl3anc 1184 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( C  x.  B )  / 
( C  x.  D
) )  =  ( B  /  D ) )
3027, 29eqtrd 2436 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( B  x.  C )  / 
( C  x.  D
) )  =  ( B  /  D ) )
3124, 30oveq12d 6058 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( ( A  x.  D )  /  ( C  x.  D ) )  +  ( ( B  x.  C )  /  ( C  x.  D )
) )  =  ( ( A  /  C
)  +  ( B  /  D ) ) )
3213, 31eqtr2d 2437 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) ) )  ->  ( ( A  /  C )  +  ( B  /  D
) )  =  ( ( ( A  x.  D )  +  ( B  x.  C ) )  /  ( C  x.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567  (class class class)co 6040   CCcc 8944   0cc0 8946    + caddc 8949    x. cmul 8951    / cdiv 9633
This theorem is referenced by:  divsubdiv  9686  divadddivi  9732  divadddivd  9790  qaddcl  10546
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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