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Theorem mulgcd 12016
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 9177 . . 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 9373 . . . . . . . 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 9380 . . . . . . 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 9380 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N )  e.  ZZ )
85, 7gcdcld 11968 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e. 
NN0 )
92nnnn0d 9228 . . . . . . 7  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  NN0 )
10 gcdcl 11966 . . . . . . . 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 9233 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e. 
NN0 )
138nn0cnd 9230 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  CC )
142nncnd 8932 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  CC )
152nnap0d 8964 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K #  0 )
1613, 14, 15divcanap2d 8748 . . . . . . 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 11963 . . . . . . . . . . . . 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 4025 . . . . . . . . . 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 11819 . . . . . . . . . . . . . 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 11819 . . . . . . . . . . . . . 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 12012 . . . . . . . . . . . . . 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 8963 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  =/=  0 )
298nn0zd 9372 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  e.  ZZ )
30 dvdsval2 11796 . . . . . . . . . . . . 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 11826 . . . . . . . . . . 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 4025 . . . . . . . . . 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 11826 . . . . . . . . . . 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 12012 . . . . . . . . . 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 9372 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  ZZ )
45 dvdscmul 11824 . . . . . . . . 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 4027 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  ||  ( K  x.  ( M  gcd  N ) ) )
49 gcddvds 11963 . . . . . . . . . 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 11824 . . . . . . . . 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 11824 . . . . . . . . 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 9372 . . . . . . . 8  |-  ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  e.  ZZ )
60 dvdsgcd 12012 . . . . . . . 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 11853 . . . . . 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 11960 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
67103adant1 1015 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
6867nn0cnd 9230 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
6968mul02d 8348 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  ( M  gcd  N ) )  =  0 )
7066, 69eqtr4id 2229 . . . . . 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 5889 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  ( 0  x.  M ) )
73 zcn 9257 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
74733ad2ant2 1019 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7574mul02d 8348 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  M
)  =  0 )
7672, 75eqtrd 2210 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M
)  =  0 )
7771oveq1d 5889 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  ( 0  x.  N ) )
78 zcn 9257 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
79783ad2ant3 1020 . . . . . . . . 9  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
8079mul02d 8348 . . . . . . . 8  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  x.  N
)  =  0 )
8177, 80eqtrd 2210 . . . . . . 7  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  =  0 )
8276, 81oveq12d 5892 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N )
)  =  ( 0  gcd  0 ) )
8371oveq1d 5889 . . . . . 6  |-  ( ( K  =  0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  ( M  gcd  N ) )  =  ( 0  x.  ( M  gcd  N
) ) )
8470, 82, 833eqtr4d 2220 . . . . 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 2148    =/= wne 2347   class class class wbr 4003  (class class class)co 5874   CCcc 7808   0cc0 7810    x. cmul 7815    / cdiv 8628   NNcn 8918   NN0cn0 9175   ZZcz 9252    || cdvds 11793    gcd cgcd 11942
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-sup 6982  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-fz 10008  df-fzo 10142  df-fl 10269  df-mod 10322  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-dvds 11794  df-gcd 11943
This theorem is referenced by:  absmulgcd  12017  mulgcdr  12018  mulgcddvds  12093  qredeu  12096  coprimeprodsq  12256  pythagtriplem4  12267  2sqlem8  14440
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