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Theorem divmuldivap 8618
Description: Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
Assertion
Ref Expression
divmuldivap  |-  ( ( ( 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 divmuldivap
StepHypRef Expression
1 3anass 977 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) )
2 3anass 977 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )
3 divclap 8584 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
4 divclap 8584 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 7890 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 287 . . . . 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 7890 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 506 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
983adantr1 1151 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1148 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulap0 8561 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
12113adantr1 1151 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
) #  0 )
13123adantl1 1148 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
14 divcanap3 8604 . . . . 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 1233 . . . 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 993 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  C  e.  CC )
1716, 3jca 304 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 993 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  D  e.  CC )
1918, 4jca 304 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 8040 . . . . . . 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 287 . . . . . 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 divcanap2 8586 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcanap2 8586 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 5870 . . . . . 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 2205 . . . . 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 5866 . . . 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 2205 . . 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 290 . 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 583 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 103    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3987  (class class class)co 5851   CCcc 7761   0cc0 7763    x. cmul 7768   # cap 8489    / cdiv 8578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579
This theorem is referenced by:  divdivdivap  8619  divcanap5  8620  divmul13ap  8621  divmul24ap  8622  divmuldivapi  8678  divmuldivapd  8738  qmulcl  9585  mulexpzap  10505  expaddzap  10509  sqdivap  10529
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