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Theorem mulgcd 12966
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 10148 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 simp1 957 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  NN )
32nnzd 10299 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
4 simp2 958 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
53, 4zmulcld 10306 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
6 simp3 959 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
73, 6zmulcld 10306 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
85, 7gcdcld 12938 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e. 
NN0 )
92nnnn0d 10199 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  NN0 )
10 gcdcl 12937 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
11103adant1 975 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e. 
NN0 )
129, 11nn0mulcld 10204 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e. 
NN0 )
138nn0cnd 10201 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  CC )
142nncnd 9941 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  CC )
152nnne0d 9969 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  =/=  0 )
1613, 14, 15divcan2d 9717 . . . . . . 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 12935 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 446 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  M
) )
2016, 19eqbrtrd 4166 . . . . . . . . . 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 12791 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  K  ||  ( K  x.  M ) )
223, 4, 21syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  M
) )
23 dvdsmul1 12791 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  N ) )
243, 6, 23syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  N
) )
25 dvdsgcd 12963 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
288nn0zd 10298 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  ZZ )
29 dvdsval2 12775 . . . . . . . . . . . . 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 1184 . . . . . . . . . . . 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 202 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K )  e.  ZZ )
32 dvdscmulr 12798 . . . . . . . . . . 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 1188 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  M )
3518simprd 450 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  N
) )
3616, 35eqbrtrd 4166 . . . . . . . . . 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 12798 . . . . . . . . . . 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 1188 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  N )
40 dvdsgcd 12963 . . . . . . . . . 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 1184 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  ( M  gcd  N ) )
4311nn0zd 10298 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  ZZ )
44 dvdscmul 12796 . . . . . . . . 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 1184 . . . . . . . 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 4168 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  ( M  gcd  N ) ) )
48 gcddvds 12935 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
49483adant1 975 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  N ) )
5049simpld 446 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  M )
51 dvdscmul 12796 . . . . . . . . 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 1184 . . . . . . . 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 450 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  N )
55 dvdscmul 12796 . . . . . . . . 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 1184 . . . . . . . 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 10298 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e.  ZZ )
59 dvdsgcd 12963 . . . . . . . 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 1184 . . . . . . 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 661 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  ||  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
62 dvdseq 12817 . . . . . 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 1185 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( K  x.  ( M  gcd  N ) ) )
64633expib 1156 . . . 4  |-  ( K  e.  NN  ->  (
( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N
) )  =  ( K  x.  ( M  gcd  N ) ) ) )
65103adant1 975 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
6665nn0cnd 10201 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
6766mul02d 9189 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  ( M  gcd  N ) )  =  0 )
68 gcd0val 12929 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
6967, 68syl6reqr 2431 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  0
)  =  ( 0  x.  ( M  gcd  N ) ) )
70 simp1 957 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  =  0 )
7170oveq1d 6028 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  ( 0  x.  M ) )
72 zcn 10212 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
73723ad2ant2 979 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7473mul02d 9189 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  M
)  =  0 )
7571, 74eqtrd 2412 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  0 )
7670oveq1d 6028 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  ( 0  x.  N ) )
77 zcn 10212 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
78773ad2ant3 980 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
7978mul02d 9189 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  N
)  =  0 )
8076, 79eqtrd 2412 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  0 )
8175, 80oveq12d 6031 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( 0  gcd  0 ) )
8270oveq1d 6028 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  =  ( 0  x.  ( M  gcd  N
) ) )
8369, 81, 823eqtr4d 2422 . . . . 5  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( K  x.  ( M  gcd  N ) ) )
84833expib 1156 . . . 4  |-  ( K  =  0  ->  (
( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N
) )  =  ( K  x.  ( M  gcd  N ) ) ) )
8564, 84jaoi 369 . . 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 188 . 2  |-  ( K  e.  NN0  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( K  x.  ( M  gcd  N ) ) ) )
87863impib 1151 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 set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146  (class class class)co 6013   CCcc 8914   0cc0 8916    x. cmul 8921    / cdiv 9602   NNcn 9925   NN0cn0 10146   ZZcz 10207    || cdivides 12772    gcd cgcd 12926
This theorem is referenced by:  absmulgcd  12967  mulgcdr  12968  mulgcddvds  13024  qredeu  13027  coprimeprodsq  13103  pythagtriplem4  13113  odadd2  15384  2sqlem8  21016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-dvds 12773  df-gcd 12927
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