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Theorem rpmulgcd2 14649
Description: If  M is relatively prime to  N, then the GCD of  K with  M  x.  N is the product of the GCDs with  M and  N respectively. (Contributed by Mario Carneiro, 2-Jul-2015.)
Assertion
Ref Expression
rpmulgcd2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  =  ( ( K  gcd  M
)  x.  ( K  gcd  N ) ) )

Proof of Theorem rpmulgcd2
StepHypRef Expression
1 simpl1 1008 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  K  e.  ZZ )
2 simpl2 1009 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  M  e.  ZZ )
3 simpl3 1010 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  N  e.  ZZ )
42, 3zmulcld 11046 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 14469 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  e.  NN0 )
61, 2gcdcld 14469 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  e.  NN0 )
71, 3gcdcld 14469 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  e.  NN0 )
86, 7nn0mulcld 10930 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  e.  NN0 )
9 mulgcddvds 14648 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )
109adantr 466 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) )
11 gcddvds 14464 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  M ) )
121, 2, 11syl2anc 665 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M )  ||  M ) )
1312simpld 460 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  ||  K )
14 gcddvds 14464 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  N ) )
151, 3, 14syl2anc 665 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N )  ||  N ) )
1615simpld 460 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  ||  K )
176nn0zd 11038 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  e.  ZZ )
187nn0zd 11038 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  e.  ZZ )
19 gcddvds 14464 . . . . . . . . . . 11  |-  ( ( ( K  gcd  M
)  e.  ZZ  /\  ( K  gcd  N )  e.  ZZ )  -> 
( ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  ( K  gcd  M )  /\  ( ( K  gcd  M )  gcd  ( K  gcd  N ) ) 
||  ( K  gcd  N ) ) )
2017, 18, 19syl2anc 665 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( K  gcd  M )  gcd  ( K  gcd  N ) )  ||  ( K  gcd  M )  /\  ( ( K  gcd  M )  gcd  ( K  gcd  N ) ) 
||  ( K  gcd  N ) ) )
2120simpld 460 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  ( K  gcd  M ) )
2212simprd 464 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  ||  M )
2317, 18gcdcld 14469 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  e.  NN0 )
2423nn0zd 11038 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  e.  ZZ )
25 dvdstr 14324 . . . . . . . . . 10  |-  ( ( ( ( K  gcd  M )  gcd  ( K  gcd  N ) )  e.  ZZ  /\  ( K  gcd  M )  e.  ZZ  /\  M  e.  ZZ )  ->  (
( ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  ( K  gcd  M )  /\  ( K  gcd  M ) 
||  M )  -> 
( ( K  gcd  M )  gcd  ( K  gcd  N ) ) 
||  M ) )
2624, 17, 2, 25syl3anc 1264 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( ( K  gcd  M
)  gcd  ( K  gcd  N ) )  ||  ( K  gcd  M )  /\  ( K  gcd  M )  ||  M )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  M
) )
2721, 22, 26mp2and 683 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  M
)
2820simprd 464 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  ( K  gcd  N ) )
2915simprd 464 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  ||  N )
30 dvdstr 14324 . . . . . . . . . 10  |-  ( ( ( ( K  gcd  M )  gcd  ( K  gcd  N ) )  e.  ZZ  /\  ( K  gcd  N )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  ( K  gcd  N )  /\  ( K  gcd  N ) 
||  N )  -> 
( ( K  gcd  M )  gcd  ( K  gcd  N ) ) 
||  N ) )
3124, 18, 3, 30syl3anc 1264 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( ( K  gcd  M
)  gcd  ( K  gcd  N ) )  ||  ( K  gcd  N )  /\  ( K  gcd  N )  ||  N )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  N
) )
3228, 29, 31mp2and 683 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  N
)
33 dvdsgcd 14498 . . . . . . . . 9  |-  ( ( ( ( K  gcd  M )  gcd  ( K  gcd  N ) )  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  M  /\  ( ( K  gcd  M )  gcd  ( K  gcd  N ) ) 
||  N )  -> 
( ( K  gcd  M )  gcd  ( K  gcd  N ) ) 
||  ( M  gcd  N ) ) )
3424, 2, 3, 33syl3anc 1264 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( ( K  gcd  M
)  gcd  ( K  gcd  N ) )  ||  M  /\  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  N
)  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  ( M  gcd  N ) ) )
3527, 32, 34mp2and 683 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  ( M  gcd  N ) )
36 simpr 462 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  gcd  N )  =  1 )
3735, 36breqtrd 4445 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  1
)
38 dvds1 14340 . . . . . . 7  |-  ( ( ( K  gcd  M
)  gcd  ( K  gcd  N ) )  e. 
NN0  ->  ( ( ( K  gcd  M )  gcd  ( K  gcd  N ) )  ||  1  <->  ( ( K  gcd  M
)  gcd  ( K  gcd  N ) )  =  1 ) )
3923, 38syl 17 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( K  gcd  M )  gcd  ( K  gcd  N ) )  ||  1  <->  ( ( K  gcd  M
)  gcd  ( K  gcd  N ) )  =  1 ) )
4037, 39mpbid 213 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  =  1 )
41 coprmdvds2 14647 . . . . 5  |-  ( ( ( ( K  gcd  M )  e.  ZZ  /\  ( K  gcd  N )  e.  ZZ  /\  K  e.  ZZ )  /\  (
( K  gcd  M
)  gcd  ( K  gcd  N ) )  =  1 )  ->  (
( ( K  gcd  M )  ||  K  /\  ( K  gcd  N ) 
||  K )  -> 
( ( K  gcd  M )  x.  ( K  gcd  N ) ) 
||  K ) )
4217, 18, 1, 40, 41syl31anc 1267 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( K  gcd  M ) 
||  K  /\  ( K  gcd  N )  ||  K )  ->  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) 
||  K ) )
4313, 16, 42mp2and 683 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  K
)
44 dvdscmul 14316 . . . . . 6  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  N  e.  ZZ  /\  ( K  gcd  M )  e.  ZZ )  ->  (
( K  gcd  N
)  ||  N  ->  ( ( K  gcd  M
)  x.  ( K  gcd  N ) ) 
||  ( ( K  gcd  M )  x.  N ) ) )
4518, 3, 17, 44syl3anc 1264 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  N )  ||  N  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  (
( K  gcd  M
)  x.  N ) ) )
46 dvdsmulc 14317 . . . . . 6  |-  ( ( ( K  gcd  M
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  ||  M  ->  ( ( K  gcd  M
)  x.  N ) 
||  ( M  x.  N ) ) )
4717, 2, 3, 46syl3anc 1264 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  ||  M  ->  ( ( K  gcd  M )  x.  N )  ||  ( M  x.  N )
) )
4817, 18zmulcld 11046 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  e.  ZZ )
4917, 3zmulcld 11046 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  N )  e.  ZZ )
50 dvdstr 14324 . . . . . 6  |-  ( ( ( ( K  gcd  M )  x.  ( K  gcd  N ) )  e.  ZZ  /\  (
( K  gcd  M
)  x.  N )  e.  ZZ  /\  ( M  x.  N )  e.  ZZ )  ->  (
( ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  (
( K  gcd  M
)  x.  N )  /\  ( ( K  gcd  M )  x.  N )  ||  ( M  x.  N )
)  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  ( M  x.  N )
) )
5148, 49, 4, 50syl3anc 1264 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( ( K  gcd  M
)  x.  ( K  gcd  N ) ) 
||  ( ( K  gcd  M )  x.  N )  /\  (
( K  gcd  M
)  x.  N ) 
||  ( M  x.  N ) )  -> 
( ( K  gcd  M )  x.  ( K  gcd  N ) ) 
||  ( M  x.  N ) ) )
5245, 47, 51syl2and 485 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( K  gcd  N ) 
||  N  /\  ( K  gcd  M )  ||  M )  ->  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) 
||  ( M  x.  N ) ) )
5329, 22, 52mp2and 683 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  ( M  x.  N )
)
54 dvdsgcd 14498 . . . 4  |-  ( ( ( ( K  gcd  M )  x.  ( K  gcd  N ) )  e.  ZZ  /\  K  e.  ZZ  /\  ( M  x.  N )  e.  ZZ )  ->  (
( ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  K  /\  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) 
||  ( M  x.  N ) )  -> 
( ( K  gcd  M )  x.  ( K  gcd  N ) ) 
||  ( K  gcd  ( M  x.  N
) ) ) )
5548, 1, 4, 54syl3anc 1264 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( ( ( K  gcd  M
)  x.  ( K  gcd  N ) ) 
||  K  /\  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) 
||  ( M  x.  N ) )  -> 
( ( K  gcd  M )  x.  ( K  gcd  N ) ) 
||  ( K  gcd  ( M  x.  N
) ) ) )
5643, 53, 55mp2and 683 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  ( K  gcd  ( M  x.  N ) ) )
57 dvdseq 14339 . 2  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  e.  NN0  /\  ( ( K  gcd  M )  x.  ( K  gcd  N ) )  e.  NN0 )  /\  ( ( K  gcd  ( M  x.  N
) )  ||  (
( K  gcd  M
)  x.  ( K  gcd  N ) )  /\  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  ( K  gcd  ( M  x.  N ) ) ) )  ->  ( K  gcd  ( M  x.  N
) )  =  ( ( K  gcd  M
)  x.  ( K  gcd  N ) ) )
585, 8, 10, 56, 57syl22anc 1265 1  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  =  ( ( K  gcd  M
)  x.  ( K  gcd  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   class class class wbr 4420  (class class class)co 6301   1c1 9540    x. cmul 9544   NN0cn0 10869   ZZcz 10937    || cdvds 14292    gcd cgcd 14455
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
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 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  df-inf 7959  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-dvds 14293  df-gcd 14456
This theorem is referenced by:  dvdsmulf1o  24109
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