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Theorem mulgcd 13039
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 10216 . . 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 10367 . . . . . . . 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 10374 . . . . . . 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 10374 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
85, 7gcdcld 13011 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e. 
NN0 )
92nnnn0d 10267 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  NN0 )
10 gcdcl 13010 . . . . . . . 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 10272 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e. 
NN0 )
138nn0cnd 10269 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  CC )
142nncnd 10009 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  CC )
152nnne0d 10037 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  =/=  0 )
1613, 14, 15divcan2d 9785 . . . . . . 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 13008 . . . . . . . . . . . . 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 4225 . . . . . . . . . 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 12864 . . . . . . . . . . . . . 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 12864 . . . . . . . . . . . . . 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 13036 . . . . . . . . . . . . . 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 10366 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  ZZ )
29 dvdsval2 12848 . . . . . . . . . . . . 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 12871 . . . . . . . . . . 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 4225 . . . . . . . . . 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 12871 . . . . . . . . . . 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 13036 . . . . . . . . . 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 10366 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  ZZ )
44 dvdscmul 12869 . . . . . . . . 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 4227 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  ( M  gcd  N ) ) )
48 gcddvds 13008 . . . . . . . . . 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 12869 . . . . . . . . 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 12869 . . . . . . . . 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 10366 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e.  ZZ )
59 dvdsgcd 13036 . . . . . . . 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 12890 . . . . . 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 10269 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
6766mul02d 9257 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  ( M  gcd  N ) )  =  0 )
68 gcd0val 13002 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
6967, 68syl6reqr 2487 . . . . . 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 6089 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  ( 0  x.  M ) )
72 zcn 10280 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
73723ad2ant2 979 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7473mul02d 9257 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  M
)  =  0 )
7571, 74eqtrd 2468 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  0 )
7670oveq1d 6089 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  ( 0  x.  N ) )
77 zcn 10280 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
78773ad2ant3 980 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
7978mul02d 9257 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  N
)  =  0 )
8076, 79eqtrd 2468 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  0 )
8175, 80oveq12d 6092 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( 0  gcd  0 ) )
8270oveq1d 6089 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  =  ( 0  x.  ( M  gcd  N
) ) )
8369, 81, 823eqtr4d 2478 . . . . 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 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4205  (class class class)co 6074   CCcc 8981   0cc0 8983    x. cmul 8988    / cdiv 9670   NNcn 9993   NN0cn0 10214   ZZcz 10275    || cdivides 12845    gcd cgcd 12999
This theorem is referenced by:  absmulgcd  13040  mulgcdr  13041  mulgcddvds  13097  qredeu  13100  coprimeprodsq  13176  pythagtriplem4  13186  odadd2  15457  2sqlem8  21149
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 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061
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 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-sup 7439  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-n0 10215  df-z 10276  df-uz 10482  df-rp 10606  df-fl 11195  df-mod 11244  df-seq 11317  df-exp 11376  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-dvds 12846  df-gcd 13000
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