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Theorem mulgcd 14060
Description: Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)
Assertion
Ref Expression
mulgcd  |-  ( ( K  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( K  x.  ( M  gcd  N ) ) )

Proof of Theorem mulgcd
StepHypRef Expression
1 elnn0 10809 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 simp1 996 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  NN )
32nnzd 10977 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
4 simp2 997 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
53, 4zmulcld 10984 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
6 simp3 998 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
73, 6zmulcld 10984 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
85, 7gcdcld 14032 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e. 
NN0 )
92nnnn0d 10864 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  NN0 )
10 gcdcl 14031 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
11103adant1 1014 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e. 
NN0 )
129, 11nn0mulcld 10869 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e. 
NN0 )
138nn0cnd 10866 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  CC )
142nncnd 10564 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  CC )
152nnne0d 10592 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  =/=  0 )
1613, 14, 15divcan2d 10334 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K ) )  =  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
17 gcddvds 14029 . . . . . . . . . . . . 13  |-  ( ( ( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ )  ->  ( ( ( K  x.  M )  gcd  ( K  x.  N ) )  ||  ( K  x.  M
)  /\  ( ( K  x.  M )  gcd  ( K  x.  N
) )  ||  ( K  x.  N )
) )
185, 7, 17syl2anc 661 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  ||  ( K  x.  M )  /\  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  N
) ) )
1918simpld 459 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  M
) )
2016, 19eqbrtrd 4473 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K ) )  ||  ( K  x.  M
) )
21 dvdsmul1 13883 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  K  ||  ( K  x.  M ) )
223, 4, 21syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  M
) )
23 dvdsmul1 13883 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  N ) )
243, 6, 23syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  N
) )
25 dvdsgcd 14057 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  ( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ )  ->  ( ( K 
||  ( K  x.  M )  /\  K  ||  ( K  x.  N
) )  ->  K  ||  ( ( K  x.  M )  gcd  ( K  x.  N )
) ) )
263, 5, 7, 25syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  ( K  x.  M )  /\  K  ||  ( K  x.  N ) )  ->  K  ||  (
( K  x.  M
)  gcd  ( K  x.  N ) ) ) )
2722, 24, 26mp2and 679 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
288nn0zd 10976 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  ZZ )
29 dvdsval2 13867 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  K  =/=  0  /\  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  ZZ )  ->  ( K  ||  ( ( K  x.  M )  gcd  ( K  x.  N
) )  <->  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K )  e.  ZZ ) )
303, 15, 28, 29syl3anc 1228 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  ||  ( ( K  x.  M )  gcd  ( K  x.  N
) )  <->  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K )  e.  ZZ ) )
3127, 30mpbid 210 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K )  e.  ZZ )
32 dvdscmulr 13890 . . . . . . . . . . 11  |-  ( ( ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  e.  ZZ  /\  M  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
) )  ||  ( K  x.  M )  <->  ( ( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  M ) )
3331, 4, 3, 15, 32syl112anc 1232 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) )  ||  ( K  x.  M )  <->  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K )  ||  M
) )
3420, 33mpbid 210 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  M )
3518simprd 463 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  N
) )
3616, 35eqbrtrd 4473 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K ) )  ||  ( K  x.  N
) )
37 dvdscmulr 13890 . . . . . . . . . . 11  |-  ( ( ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( K  x.  ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
) )  ||  ( K  x.  N )  <->  ( ( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  N ) )
3831, 6, 3, 15, 37syl112anc 1232 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) )  ||  ( K  x.  N )  <->  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K )  ||  N
) )
3936, 38mpbid 210 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  N )
40 dvdsgcd 14057 . . . . . . . . . 10  |-  ( ( ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( ( ( K  x.  M )  gcd  ( K  x.  N ) )  /  K )  ||  M  /\  ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  ||  N )  ->  ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  ||  ( M  gcd  N ) ) )
4131, 4, 6, 40syl3anc 1228 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( ( ( K  x.  M )  gcd  ( K  x.  N ) )  /  K )  ||  M  /\  ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  ||  N )  ->  ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  ||  ( M  gcd  N ) ) )
4234, 39, 41mp2and 679 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  ( M  gcd  N ) )
4311nn0zd 10976 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  ZZ )
44 dvdscmul 13888 . . . . . . . . 9  |-  ( ( ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  e.  ZZ  /\  ( M  gcd  N )  e.  ZZ  /\  K  e.  ZZ )  ->  (
( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  ||  ( M  gcd  N )  ->  ( K  x.  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K ) )  ||  ( K  x.  ( M  gcd  N ) ) ) )
4531, 43, 3, 44syl3anc 1228 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( ( K  x.  M )  gcd  ( K  x.  N
) )  /  K
)  ||  ( M  gcd  N )  ->  ( K  x.  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K ) )  ||  ( K  x.  ( M  gcd  N ) ) ) )
4642, 45mpd 15 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K ) )  ||  ( K  x.  ( M  gcd  N ) ) )
4716, 46eqbrtrrd 4475 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  ( M  gcd  N ) ) )
48 gcddvds 14029 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
49483adant1 1014 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  N ) )
5049simpld 459 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  M )
51 dvdscmul 13888 . . . . . . . . 9  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  ->  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  M ) ) )
5243, 4, 3, 51syl3anc 1228 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  ->  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  M ) ) )
5350, 52mpd 15 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  ||  ( K  x.  M
) )
5449simprd 463 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  N )
55 dvdscmul 13888 . . . . . . . . 9  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M  gcd  N
)  ||  N  ->  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  N ) ) )
5643, 6, 3, 55syl3anc 1228 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  N  ->  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  N ) ) )
5754, 56mpd 15 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  ||  ( K  x.  N
) )
5812nn0zd 10976 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e.  ZZ )
59 dvdsgcd 14057 . . . . . . . 8  |-  ( ( ( K  x.  ( M  gcd  N ) )  e.  ZZ  /\  ( K  x.  M )  e.  ZZ  /\  ( K  x.  N )  e.  ZZ )  ->  (
( ( K  x.  ( M  gcd  N ) )  ||  ( K  x.  M )  /\  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  N ) )  -> 
( K  x.  ( M  gcd  N ) ) 
||  ( ( K  x.  M )  gcd  ( K  x.  N
) ) ) )
6058, 5, 7, 59syl3anc 1228 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  ( M  gcd  N ) )  ||  ( K  x.  M )  /\  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  N ) )  -> 
( K  x.  ( M  gcd  N ) ) 
||  ( ( K  x.  M )  gcd  ( K  x.  N
) ) ) )
6153, 57, 60mp2and 679 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  ||  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
62 dvdseq 13909 . . . . . 6  |-  ( ( ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  e.  NN0  /\  ( K  x.  ( M  gcd  N ) )  e.  NN0 )  /\  ( ( ( K  x.  M )  gcd  ( K  x.  N
) )  ||  ( K  x.  ( M  gcd  N ) )  /\  ( K  x.  ( M  gcd  N ) ) 
||  ( ( K  x.  M )  gcd  ( K  x.  N
) ) ) )  ->  ( ( K  x.  M )  gcd  ( K  x.  N
) )  =  ( K  x.  ( M  gcd  N ) ) )
638, 12, 47, 61, 62syl22anc 1229 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( K  x.  ( M  gcd  N ) ) )
64633expib 1199 . . . 4  |-  ( K  e.  NN  ->  (
( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N
) )  =  ( K  x.  ( M  gcd  N ) ) ) )
65103adant1 1014 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
6665nn0cnd 10866 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
6766mul02d 9789 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  ( M  gcd  N ) )  =  0 )
68 gcd0val 14023 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
6967, 68syl6reqr 2527 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  0
)  =  ( 0  x.  ( M  gcd  N ) ) )
70 simp1 996 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  =  0 )
7170oveq1d 6310 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  ( 0  x.  M ) )
72 zcn 10881 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
73723ad2ant2 1018 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7473mul02d 9789 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  M
)  =  0 )
7571, 74eqtrd 2508 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  0 )
7670oveq1d 6310 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  ( 0  x.  N ) )
77 zcn 10881 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
78773ad2ant3 1019 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
7978mul02d 9789 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  N
)  =  0 )
8076, 79eqtrd 2508 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  0 )
8175, 80oveq12d 6313 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( 0  gcd  0 ) )
8270oveq1d 6310 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  =  ( 0  x.  ( M  gcd  N
) ) )
8369, 81, 823eqtr4d 2518 . . . . 5  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( K  x.  ( M  gcd  N ) ) )
84833expib 1199 . . . 4  |-  ( K  =  0  ->  (
( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N
) )  =  ( K  x.  ( M  gcd  N ) ) ) )
8564, 84jaoi 379 . . 3  |-  ( ( K  e.  NN  \/  K  =  0 )  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( K  x.  ( M  gcd  N ) ) ) )
861, 85sylbi 195 . 2  |-  ( K  e.  NN0  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( K  x.  ( M  gcd  N ) ) ) )
87863impib 1194 1  |-  ( ( K  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( K  x.  ( M  gcd  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453  (class class class)co 6295   CCcc 9502   0cc0 9504    x. cmul 9509    / cdiv 10218   NNcn 10548   NN0cn0 10807   ZZcz 10876    || cdivides 13864    gcd cgcd 14020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021
This theorem is referenced by:  absmulgcd  14061  mulgcdr  14062  mulgcddvds  14121  qredeu  14124  coprimeprodsq  14209  pythagtriplem4  14219  odadd2  16728  2sqlem8  23513
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