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Theorem mulgcd 11982
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 9149 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 simp1 997 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  NN )
32nnzd 9345 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
4 simp2 998 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
53, 4zmulcld 9352 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
6 simp3 999 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
73, 6zmulcld 9352 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
85, 7gcdcld 11934 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e. 
NN0 )
92nnnn0d 9200 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  NN0 )
10 gcdcl 11932 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
11103adant1 1015 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e. 
NN0 )
129, 11nn0mulcld 9205 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e. 
NN0 )
138nn0cnd 9202 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  CC )
142nncnd 8904 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  CC )
152nnap0d 8936 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K #  0 )
1613, 14, 15divcanap2d 8721 . . . . . . 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 11929 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 112 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  M
) )
2016, 19eqbrtrd 4020 . . . . . . . . . 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 11786 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  K  ||  ( K  x.  M ) )
223, 4, 21syl2anc 411 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  M
) )
23 dvdsmul1 11786 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  N ) )
243, 6, 23syl2anc 411 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( K  x.  N
) )
25 dvdsgcd 11978 . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . 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 433 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  ||  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
282nnne0d 8935 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  =/=  0 )
298nn0zd 9344 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  ZZ )
30 dvdsval2 11763 . . . . . . . . . . . . 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 ) )
313, 28, 29, 30syl3anc 1238 . . . . . . . . . . . 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 ) )
3227, 31mpbid 147 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K )  e.  ZZ )
33 dvdscmulr 11793 . . . . . . . . . . 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 ) )
3432, 4, 3, 28, 33syl112anc 1242 . . . . . . . . . 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
) )
3520, 34mpbid 147 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  M )
3618simprd 114 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  N
) )
3716, 36eqbrtrd 4020 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( (
( K  x.  M
)  gcd  ( K  x.  N ) )  /  K ) )  ||  ( K  x.  N
) )
38 dvdscmulr 11793 . . . . . . . . . . 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 ) )
3932, 6, 3, 28, 38syl112anc 1242 . . . . . . . . . 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
) )
4037, 39mpbid 147 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  N )
41 dvdsgcd 11978 . . . . . . . . . 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 ) ) )
4232, 4, 6, 41syl3anc 1238 . . . . . . . . 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 ) ) )
4335, 40, 42mp2and 433 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  x.  M )  gcd  ( K  x.  N )
)  /  K ) 
||  ( M  gcd  N ) )
4411nn0zd 9344 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  ZZ )
45 dvdscmul 11791 . . . . . . . . 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 ) ) ) )
4632, 44, 3, 45syl3anc 1238 . . . . . . . 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 ) ) ) )
4743, 46mpd 13 . . . . . . 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 ) ) )
4816, 47eqbrtrrd 4022 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  ( M  gcd  N ) ) )
49 gcddvds 11929 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
50493adant1 1015 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  N ) )
5150simpld 112 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  M )
52 dvdscmul 11791 . . . . . . . . 9  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  ->  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  M ) ) )
5344, 4, 3, 52syl3anc 1238 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  ->  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  M ) ) )
5451, 53mpd 13 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  ||  ( K  x.  M
) )
5550simprd 114 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  N )
56 dvdscmul 11791 . . . . . . . . 9  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M  gcd  N
)  ||  N  ->  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  N ) ) )
5744, 6, 3, 56syl3anc 1238 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  N  ->  ( K  x.  ( M  gcd  N ) ) 
||  ( K  x.  N ) ) )
5855, 57mpd 13 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  ||  ( K  x.  N
) )
5912nn0zd 9344 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e.  ZZ )
60 dvdsgcd 11978 . . . . . . . 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
) ) ) )
6159, 5, 7, 60syl3anc 1238 . . . . . . 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
) ) ) )
6254, 58, 61mp2and 433 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  ||  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
63 dvdseq 11819 . . . . . 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 ) ) )
648, 12, 48, 62, 63syl22anc 1239 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( K  x.  ( M  gcd  N ) ) )
65643expib 1206 . . . 4  |-  ( K  e.  NN  ->  (
( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N
) )  =  ( K  x.  ( M  gcd  N ) ) ) )
66 gcd0val 11926 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
67103adant1 1015 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
6867nn0cnd 9202 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
6968mul02d 8323 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  ( M  gcd  N ) )  =  0 )
7066, 69eqtr4id 2227 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  0
)  =  ( 0  x.  ( M  gcd  N ) ) )
71 simp1 997 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  =  0 )
7271oveq1d 5880 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  ( 0  x.  M ) )
73 zcn 9229 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
74733ad2ant2 1019 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7574mul02d 8323 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  M
)  =  0 )
7672, 75eqtrd 2208 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  0 )
7771oveq1d 5880 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  ( 0  x.  N ) )
78 zcn 9229 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
79783ad2ant3 1020 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
8079mul02d 8323 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  N
)  =  0 )
8177, 80eqtrd 2208 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  0 )
8276, 81oveq12d 5883 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( 0  gcd  0 ) )
8371oveq1d 5880 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  =  ( 0  x.  ( M  gcd  N
) ) )
8470, 82, 833eqtr4d 2218 . . . . 5  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( K  x.  ( M  gcd  N ) ) )
85843expib 1206 . . . 4  |-  ( K  =  0  ->  (
( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N
) )  =  ( K  x.  ( M  gcd  N ) ) ) )
8665, 85jaoi 716 . . 3  |-  ( ( K  e.  NN  \/  K  =  0 )  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( K  x.  ( M  gcd  N ) ) ) )
871, 86sylbi 121 . 2  |-  ( K  e.  NN0  ->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( K  x.  ( M  gcd  N ) ) ) )
88873impib 1201 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    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2146    =/= wne 2345   class class class wbr 3998  (class class class)co 5865   CCcc 7784   0cc0 7786    x. cmul 7791    / cdiv 8601   NNcn 8890   NN0cn0 9147   ZZcz 9224    || cdvds 11760    gcd cgcd 11908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909
This theorem is referenced by:  absmulgcd  11983  mulgcdr  11984  mulgcddvds  12059  qredeu  12062  coprimeprodsq  12222  pythagtriplem4  12233  2sqlem8  14010
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