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Theorem mulgcddvds 11169
Description: One half of rpmulgcd2 11170, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)
Assertion
Ref Expression
mulgcddvds  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )

Proof of Theorem mulgcddvds
StepHypRef Expression
1 simp1 943 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
2 simp2 944 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
3 simp3 945 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
42, 3zmulcld 8844 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 11053 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e. 
NN0 )
65nn0zd 8836 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e.  ZZ )
7 dvds0 10904 . . . . 5  |-  ( ( K  gcd  ( M  x.  N ) )  e.  ZZ  ->  ( K  gcd  ( M  x.  N ) )  ||  0 )
86, 7syl 14 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  0 )
98adantr 270 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  0
)
10 oveq2 5642 . . . 4  |-  ( ( K  gcd  N )  =  0  ->  (
( K  gcd  M
)  x.  ( K  gcd  N ) )  =  ( ( K  gcd  M )  x.  0 ) )
111, 2gcdcld 11053 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e. 
NN0 )
1211nn0cnd 8698 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  CC )
1312mul01d 7850 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  x.  0 )  =  0 )
1410, 13sylan9eqr 2142 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  0 )
159, 14breqtrrd 3863 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) )
166adantr 270 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  ZZ )
1716zcnd 8839 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  CC )
181, 3gcdcld 11053 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e. 
NN0 )
1918nn0zd 8836 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  ZZ )
2019adantr 270 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  ZZ )
2120zcnd 8839 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  CC )
22 0zd 8732 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  e.  ZZ )
23 zapne 8791 . . . . . 6  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( K  gcd  N ) #  0  <->  ( K  gcd  N )  =/=  0
) )
2419, 22, 23syl2anc 403 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
) #  0  <->  ( K  gcd  N )  =/=  0
) )
2524biimpar 291 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N ) #  0 )
2617, 21, 25divcanap1d 8231 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) )  =  ( K  gcd  ( M  x.  N
) ) )
27 gcddvds 11048 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( ( K  gcd  ( M  x.  N ) )  ||  K  /\  ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  N )
) )
281, 4, 27syl2anc 403 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  /\  ( K  gcd  ( M  x.  N ) ) 
||  ( M  x.  N ) ) )
2928simpld 110 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  K )
306, 1, 19, 29dvdsmultr1d 10928 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( K  x.  ( K  gcd  N ) ) )
3130adantr 270 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  ( K  x.  ( K  gcd  N ) ) )
3226, 31eqbrtrd 3857 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( K  x.  ( K  gcd  N ) ) )
33 gcddvds 11048 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  N ) )
341, 3, 33syl2anc 403 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  K  /\  ( K  gcd  N ) 
||  N ) )
3534simpld 110 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  K )
3634simprd 112 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  N )
37 dvdsmultr2 10929 . . . . . . . . . . . 12  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  N  ->  ( K  gcd  N ) 
||  ( M  x.  N ) ) )
3819, 2, 3, 37syl3anc 1174 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  N  ->  ( K  gcd  N ) 
||  ( M  x.  N ) ) )
3936, 38mpd 13 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  ( M  x.  N
) )
40 dvdsgcd 11094 . . . . . . . . . . 11  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  K  e.  ZZ  /\  ( M  x.  N )  e.  ZZ )  ->  (
( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  ( M  x.  N ) )  -> 
( K  gcd  N
)  ||  ( K  gcd  ( M  x.  N
) ) ) )
4119, 1, 4, 40syl3anc 1174 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  ( M  x.  N ) )  -> 
( K  gcd  N
)  ||  ( K  gcd  ( M  x.  N
) ) ) )
4235, 39, 41mp2and 424 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) ) )
4342adantr 270 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) ) )
44 simpr 108 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  =/=  0 )
45 dvdsval2 10892 . . . . . . . . 9  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  ( K  gcd  N )  =/=  0  /\  ( K  gcd  ( M  x.  N ) )  e.  ZZ )  ->  (
( K  gcd  N
)  ||  ( K  gcd  ( M  x.  N
) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  e.  ZZ ) )
4620, 44, 16, 45syl3anc 1174 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) )  <-> 
( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ ) )
4743, 46mpbid 145 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  e.  ZZ )
481adantr 270 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  K  e.  ZZ )
49 dvdsmulcr 10919 . . . . . . 7  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ  /\  K  e.  ZZ  /\  ( ( K  gcd  N )  e.  ZZ  /\  ( K  gcd  N )  =/=  0 ) )  -> 
( ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( K  x.  ( K  gcd  N ) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K
) )
5047, 48, 20, 44, 49syl112anc 1178 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( ( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) )  ||  ( K  x.  ( K  gcd  N ) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K
) )
5132, 50mpbid 145 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K
)
52 nn0abscl 10483 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
532, 52syl 14 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  M )  e. 
NN0 )
5453nn0zd 8836 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  M )  e.  ZZ )
55 dvdsmultr2 10929 . . . . . . . . . . . . 13  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  ( abs `  M )  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  ->  ( K  gcd  ( M  x.  N ) ) 
||  ( ( abs `  M )  x.  K
) ) )
566, 54, 1, 55syl3anc 1174 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  ->  ( K  gcd  ( M  x.  N ) ) 
||  ( ( abs `  M )  x.  K
) ) )
5729, 56mpd 13 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  K ) )
5828simprd 112 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  N
) )
59 iddvds 10902 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  ||  M )
602, 59syl 14 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  M )
61 dvdsabsb 10908 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M  ||  M  <->  M 
||  ( abs `  M
) ) )
622, 2, 61syl2anc 403 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  M  <->  M  ||  ( abs `  M ) ) )
6360, 62mpbid 145 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( abs `  M
) )
64 dvdsmulc 10917 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  ( abs `  M )  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( abs `  M
)  ->  ( M  x.  N )  ||  (
( abs `  M
)  x.  N ) ) )
652, 54, 3, 64syl3anc 1174 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( abs `  M
)  ->  ( M  x.  N )  ||  (
( abs `  M
)  x.  N ) ) )
6663, 65mpd 13 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  ||  ( ( abs `  M
)  x.  N ) )
6754, 3zmulcld 8844 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  N )  e.  ZZ )
68 dvdstr 10926 . . . . . . . . . . . . 13  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ  /\  ( ( abs `  M
)  x.  N )  e.  ZZ )  -> 
( ( ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  N
)  /\  ( M  x.  N )  ||  (
( abs `  M
)  x.  N ) )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( abs `  M
)  x.  N ) ) )
696, 4, 67, 68syl3anc 1174 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  N )  /\  ( M  x.  N
)  ||  ( ( abs `  M )  x.  N ) )  -> 
( K  gcd  ( M  x.  N )
)  ||  ( ( abs `  M )  x.  N ) ) )
7058, 66, 69mp2and 424 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  N ) )
7154, 1zmulcld 8844 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  K )  e.  ZZ )
72 dvdsgcd 11094 . . . . . . . . . . . 12  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  ( ( abs `  M
)  x.  K )  e.  ZZ  /\  (
( abs `  M
)  x.  N )  e.  ZZ )  -> 
( ( ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  K )  /\  ( K  gcd  ( M  x.  N
) )  ||  (
( abs `  M
)  x.  N ) )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) ) ) )
736, 71, 67, 72syl3anc 1174 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  ( M  x.  N
) )  ||  (
( abs `  M
)  x.  K )  /\  ( K  gcd  ( M  x.  N
) )  ||  (
( abs `  M
)  x.  N ) )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) ) ) )
7457, 70, 73mp2and 424 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( ( abs `  M )  x.  K
)  gcd  ( ( abs `  M )  x.  N ) ) )
7518nn0red 8697 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  RR )
7618nn0ge0d 8699 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  <_  ( K  gcd  N
) )
7775, 76absidd 10565 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( K  gcd  N ) )  =  ( K  gcd  N ) )
7877oveq2d 5650 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  ( abs `  ( K  gcd  N
) ) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
792zcnd 8839 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
8018nn0cnd 8698 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  CC )
8179, 80absmuld 10592 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  ( K  gcd  N ) ) )  =  ( ( abs `  M
)  x.  ( abs `  ( K  gcd  N
) ) ) )
82 mulgcd 11098 . . . . . . . . . . . 12  |-  ( ( ( abs `  M
)  e.  NN0  /\  K  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
8353, 1, 3, 82syl3anc 1174 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
8478, 81, 833eqtr4d 2130 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  ( K  gcd  N ) ) )  =  ( ( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) ) )
8574, 84breqtrrd 3863 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) )
862, 19zmulcld 8844 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  ( K  gcd  N ) )  e.  ZZ )
87 dvdsabsb 10908 . . . . . . . . . 10  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  ( M  x.  ( K  gcd  N ) )  e.  ZZ )  -> 
( ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  ( K  gcd  N ) )  <->  ( K  gcd  ( M  x.  N
) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) ) )
886, 86, 87syl2anc 403 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  ( M  x.  ( K  gcd  N
) )  <->  ( K  gcd  ( M  x.  N
) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) ) )
8985, 88mpbird 165 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  ( K  gcd  N ) ) )
9089adantr 270 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  ( K  gcd  N ) ) )
9126, 90eqbrtrd 3857 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( M  x.  ( K  gcd  N ) ) )
922adantr 270 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  M  e.  ZZ )
93 dvdsmulcr 10919 . . . . . . 7  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ  /\  M  e.  ZZ  /\  ( ( K  gcd  N )  e.  ZZ  /\  ( K  gcd  N )  =/=  0 ) )  -> 
( ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( M  x.  ( K  gcd  N ) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  M
) )
9447, 92, 20, 44, 93syl112anc 1178 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( ( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) )  ||  ( M  x.  ( K  gcd  N ) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  M
) )
9591, 94mpbid 145 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  M
)
96 dvdsgcd 11094 . . . . . 6  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K  /\  ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) ) 
||  M )  -> 
( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) ) 
||  ( K  gcd  M ) ) )
9747, 48, 92, 96syl3anc 1174 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( ( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) ) 
||  K  /\  (
( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) ) 
||  M )  -> 
( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) ) 
||  ( K  gcd  M ) ) )
9851, 95, 97mp2and 424 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  ( K  gcd  M ) )
9911nn0zd 8836 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  ZZ )
10099adantr 270 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  M )  e.  ZZ )
101 dvdsmulc 10917 . . . . 5  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ  /\  ( K  gcd  M )  e.  ZZ  /\  ( K  gcd  N )  e.  ZZ )  ->  (
( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) ) 
||  ( K  gcd  M )  ->  ( (
( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) )  ||  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) ) )
10247, 100, 20, 101syl3anc 1174 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  ||  ( K  gcd  M )  -> 
( ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  x.  ( K  gcd  N ) ) 
||  ( ( K  gcd  M )  x.  ( K  gcd  N
) ) ) )
10398, 102mpd 13 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( ( K  gcd  M )  x.  ( K  gcd  N
) ) )
10426, 103eqbrtrrd 3859 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) )
105 zdceq 8792 . . . 4  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  0  e.  ZZ )  -> DECID  ( K  gcd  N )  =  0 )
10619, 22, 105syl2anc 403 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( K  gcd  N
)  =  0 )
107 exmiddc 782 . . . 4  |-  (DECID  ( K  gcd  N )  =  0  ->  ( ( K  gcd  N )  =  0  \/  -.  ( K  gcd  N )  =  0 ) )
108 df-ne 2256 . . . . 5  |-  ( ( K  gcd  N )  =/=  0  <->  -.  ( K  gcd  N )  =  0 )
109108orbi2i 714 . . . 4  |-  ( ( ( K  gcd  N
)  =  0  \/  ( K  gcd  N
)  =/=  0 )  <-> 
( ( K  gcd  N )  =  0  \/ 
-.  ( K  gcd  N )  =  0 ) )
110107, 109sylibr 132 . . 3  |-  (DECID  ( K  gcd  N )  =  0  ->  ( ( K  gcd  N )  =  0  \/  ( K  gcd  N )  =/=  0 ) )
111106, 110syl 14 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  =  0  \/  ( K  gcd  N
)  =/=  0 ) )
11215, 104, 111mpjaodan 747 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   0cc0 7329    x. cmul 7334   # cap 8034    / cdiv 8113   NN0cn0 8643   ZZcz 8720   abscabs 10395    || cdvds 10889    gcd cgcd 11031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10890  df-gcd 11032
This theorem is referenced by:  rpmulgcd2  11170  rpmul  11173
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