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Theorem rpmulgcd2 13105
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 960 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  K  e.  ZZ )
2 simpl2 961 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  M  e.  ZZ )
3 simpl3 962 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  N  e.  ZZ )
42, 3zmulcld 10381 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 13018 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  e.  NN0 )
61, 2gcdcld 13018 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  e.  NN0 )
71, 3gcdcld 13018 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  e.  NN0 )
86, 7nn0mulcld 10279 . 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 13104 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )
109adantr 452 . 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 13015 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M ) 
||  M ) )
121, 2, 11syl2anc 643 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  ||  K  /\  ( K  gcd  M )  ||  M ) )
1312simpld 446 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  ||  K )
14 gcddvds 13015 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  N ) )
151, 3, 14syl2anc 643 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N )  ||  N ) )
1615simpld 446 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  ||  K )
176nn0zd 10373 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  e.  ZZ )
187nn0zd 10373 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  e.  ZZ )
19 gcddvds 13015 . . . . . . . . . . 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 643 . . . . . . . . . 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 446 . . . . . . . . 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 450 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  M )  ||  M )
2317, 18gcdcld 13018 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  e.  NN0 )
2423nn0zd 10373 . . . . . . . . . 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 12883 . . . . . . . . . 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 1184 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  M
)
2820simprd 450 . . . . . . . . 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 450 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( K  gcd  N )  ||  N )
30 dvdstr 12883 . . . . . . . . . 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 1184 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  N
)
33 dvdsgcd 13043 . . . . . . . . 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 1184 . . . . . . . 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 661 . . . . . . 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 448 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( M  gcd  N )  =  1 )
3735, 36breqtrd 4236 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  ||  1
)
38 dvds1 12898 . . . . . . 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 202 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  gcd  ( K  gcd  N
) )  =  1 )
41 coprmdvds2 13103 . . . . 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 1187 . . . 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 661 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  ||  K
)
44 dvdscmul 12876 . . . . . 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 1184 . . . . 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 12877 . . . . . 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 1184 . . . . 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 10381 . . . . . 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 10381 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  gcd  N
)  =  1 )  ->  ( ( K  gcd  M )  x.  N )  e.  ZZ )
50 dvdstr 12883 . . . . . 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 1184 . . . . 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 470 . . . 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 661 . . 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 13043 . . . 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 1184 . . 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 661 . 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 12897 . 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 1185 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   1c1 8991    x. cmul 8995   NN0cn0 10221   ZZcz 10282    || cdivides 12852    gcd cgcd 13006
This theorem is referenced by:  dvdsmulf1o  20979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007
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