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Theorem rpmulgcd2 14258
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 999 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  K  e.  ZZ )
2 simpl2 1000 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  M  e.  ZZ )
3 simpl3 1001 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  N  e.  ZZ )
42, 3zmulcld 10996 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 14168 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  e.  NN0 )
61, 2gcdcld 14168 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  e.  NN0 )
71, 3gcdcld 14168 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  e.  NN0 )
86, 7nn0mulcld 10878 . 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 14257 . . 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 14165 . . . . . 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 14165 . . . . . 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 10988 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  e.  ZZ )
187nn0zd 10988 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  e.  ZZ )
19 gcddvds 14165 . . . . . . . . . . 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 14168 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  e.  NN0 )
2423nn0zd 10988 . . . . . . . . . 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 14030 . . . . . . . . . 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 1228 . . . . . . . . 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 14030 . . . . . . . . . 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 1228 . . . . . . . . 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 14193 . . . . . . . . 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 1228 . . . . . . . 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 4480 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  1
)
38 dvds1 14046 . . . . . . 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 14256 . . . . 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 1231 . . . 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 14022 . . . . . 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 1228 . . . . 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 14023 . . . . . 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 1228 . . . . 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 10996 . . . . . 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 10996 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  N )  e.  ZZ )
50 dvdstr 14030 . . . . . 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 1228 . . . . 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 14193 . . . 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 1228 . . 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 14045 . 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 1229 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 973    = wceq 1395    e. wcel 1819   class class class wbr 4456  (class class class)co 6296   1c1 9510    x. cmul 9514   NN0cn0 10816   ZZcz 10885    || cdvds 13998    gcd cgcd 14156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157
This theorem is referenced by:  dvdsmulf1o  23596
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