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Theorem rpmulgcd2 13791
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 991 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  K  e.  ZZ )
2 simpl2 992 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  M  e.  ZZ )
3 simpl3 993 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  N  e.  ZZ )
42, 3zmulcld 10753 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 13702 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  e.  NN0 )
61, 2gcdcld 13702 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  e.  NN0 )
71, 3gcdcld 13702 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  e.  NN0 )
86, 7nn0mulcld 10641 . 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 13790 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )
109adantr 465 . 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 13699 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  M ) )
121, 2, 11syl2anc 661 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M )  ||  M ) )
1312simpld 459 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  ||  K )
14 gcddvds 13699 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  N ) )
151, 3, 14syl2anc 661 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N )  ||  N ) )
1615simpld 459 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  ||  K )
176nn0zd 10745 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  e.  ZZ )
187nn0zd 10745 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  e.  ZZ )
19 gcddvds 13699 . . . . . . . . . . 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 661 . . . . . . . . . 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 459 . . . . . . . . 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 463 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  ||  M )
2317, 18gcdcld 13702 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  e.  NN0 )
2423nn0zd 10745 . . . . . . . . . 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 13566 . . . . . . . . . 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 1218 . . . . . . . . 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 679 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  M
)
2820simprd 463 . . . . . . . . 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 463 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  ||  N )
30 dvdstr 13566 . . . . . . . . . 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 1218 . . . . . . . . 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 679 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  N
)
33 dvdsgcd 13727 . . . . . . . . 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 1218 . . . . . . . 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 679 . . . . . . 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 461 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  gcd  N )  =  1 )
3735, 36breqtrd 4316 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  1
)
38 dvds1 13581 . . . . . . 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 16 . . . . . 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 210 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  =  1 )
41 coprmdvds2 13789 . . . . 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 1221 . . . 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 679 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  K
)
44 dvdscmul 13559 . . . . . 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 1218 . . . . 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 13560 . . . . . 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 1218 . . . . 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 10753 . . . . . 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 10753 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  N )  e.  ZZ )
50 dvdstr 13566 . . . . . 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 1218 . . . . 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 483 . . . 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 679 . . 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 13727 . . . 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 1218 . . 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 679 . 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 13580 . 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 1219 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292  (class class class)co 6091   1c1 9283    x. cmul 9287   NN0cn0 10579   ZZcz 10646    || cdivides 13535    gcd cgcd 13690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691
This theorem is referenced by:  dvdsmulf1o  22534
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