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Theorem gcdaddm 11661
Description: Adding a multiple of one operand of the  gcd operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.)
Assertion
Ref Expression
gcdaddm  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  ( M  gcd  ( N  +  ( K  x.  M ) ) ) )

Proof of Theorem gcdaddm
StepHypRef Expression
1 gcddvds 11641 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
213adant1 999 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  N ) )
32simpld 111 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  M )
4 simp1 981 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
5 1zzd 9074 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  1  e.  ZZ )
6 gcdcl 11644 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
763adant1 999 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e. 
NN0 )
87nn0zd 9164 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  ZZ )
9 simp2 982 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
10 simp3 983 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
11 dvds2ln 11515 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  1  e.  ZZ )  /\  ( ( M  gcd  N )  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N )  ->  ( M  gcd  N )  ||  ( ( K  x.  M )  +  ( 1  x.  N ) ) ) )
124, 5, 8, 9, 10, 11syl23anc 1223 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N )  -> 
( M  gcd  N
)  ||  ( ( K  x.  M )  +  ( 1  x.  N ) ) ) )
132, 12mpd 13 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  ( ( K  x.  M )  +  ( 1  x.  N ) ) )
1410zcnd 9167 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
1514mulid2d 7777 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
1  x.  N )  =  N )
1615oveq2d 5783 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  ( 1  x.  N ) )  =  ( ( K  x.  M )  +  N ) )
1713, 16breqtrd 3949 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  ( ( K  x.  M )  +  N
) )
183, 17jca 304 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  ( ( K  x.  M )  +  N ) ) )
194, 9zmulcld 9172 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
2019, 10zaddcld 9170 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  N )  e.  ZZ )
21 dvdslegcd 11642 . . . . . . . 8  |-  ( ( ( ( M  gcd  N )  e.  ZZ  /\  M  e.  ZZ  /\  (
( K  x.  M
)  +  N )  e.  ZZ )  /\  -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 ) )  ->  (
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  ( ( K  x.  M )  +  N ) )  -> 
( M  gcd  N
)  <_  ( M  gcd  ( ( K  x.  M )  +  N
) ) ) )
2221ex 114 . . . . . . 7  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  (
( K  x.  M
)  +  N )  e.  ZZ )  -> 
( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  -> 
( ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  ( ( K  x.  M )  +  N ) )  ->  ( M  gcd  N )  <_  ( M  gcd  ( ( K  x.  M )  +  N
) ) ) ) )
238, 9, 20, 22syl3anc 1216 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  ->  ( (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  ( ( K  x.  M )  +  N ) )  -> 
( M  gcd  N
)  <_  ( M  gcd  ( ( K  x.  M )  +  N
) ) ) ) )
2418, 23mpid 42 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  ->  ( M  gcd  N )  <_  ( M  gcd  ( ( K  x.  M )  +  N ) ) ) )
25 gcddvds 11641 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( ( K  x.  M )  +  N
)  e.  ZZ )  ->  ( ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N
) )  ||  (
( K  x.  M
)  +  N ) ) )
269, 20, 25syl2anc 408 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  (
( K  x.  M
)  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  ( ( K  x.  M )  +  N ) ) )
2726simpld 111 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  M )
284znegcld 9168 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  -u K  e.  ZZ )
299, 20gcdcld 11646 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  e. 
NN0 )
3029nn0zd 9164 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  e.  ZZ )
31 dvds2ln 11515 . . . . . . . . . 10  |-  ( ( ( -u K  e.  ZZ  /\  1  e.  ZZ )  /\  (
( M  gcd  (
( K  x.  M
)  +  N ) )  e.  ZZ  /\  M  e.  ZZ  /\  (
( K  x.  M
)  +  N )  e.  ZZ ) )  ->  ( ( ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  ( ( K  x.  M )  +  N
) )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  ( ( -u K  x.  M )  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) ) ) )
3228, 5, 30, 9, 20, 31syl23anc 1223 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( M  gcd  ( ( K  x.  M )  +  N
) )  ||  M  /\  ( M  gcd  (
( K  x.  M
)  +  N ) )  ||  ( ( K  x.  M )  +  N ) )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  ||  (
( -u K  x.  M
)  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) ) ) )
3326, 32mpd 13 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  ( ( -u K  x.  M )  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) ) )
344zcnd 9167 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  CC )
359zcnd 9167 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
3634, 35mulneg1d 8166 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u K  x.  M )  =  -u ( K  x.  M ) )
3720zcnd 9167 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  N )  e.  CC )
3837mulid2d 7777 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
1  x.  ( ( K  x.  M )  +  N ) )  =  ( ( K  x.  M )  +  N ) )
3936, 38oveq12d 5785 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -u K  x.  M
)  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) )  =  ( -u ( K  x.  M )  +  ( ( K  x.  M )  +  N ) ) )
4034, 35mulcld 7779 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  CC )
4140negcld 8053 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  -u ( K  x.  M )  e.  CC )
4240, 41addcomd 7906 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  -u ( K  x.  M )
)  =  ( -u ( K  x.  M
)  +  ( K  x.  M ) ) )
4340negidd 8056 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  -u ( K  x.  M )
)  =  0 )
4442, 43eqtr3d 2172 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u ( K  x.  M
)  +  ( K  x.  M ) )  =  0 )
4544oveq1d 5782 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -u ( K  x.  M )  +  ( K  x.  M ) )  +  N )  =  ( 0  +  N ) )
4641, 40, 14addassd 7781 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -u ( K  x.  M )  +  ( K  x.  M ) )  +  N )  =  ( -u ( K  x.  M )  +  ( ( K  x.  M )  +  N ) ) )
4714addid2d 7905 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
0  +  N )  =  N )
4845, 46, 473eqtr3d 2178 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u ( K  x.  M
)  +  ( ( K  x.  M )  +  N ) )  =  N )
4939, 48eqtrd 2170 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -u K  x.  M
)  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) )  =  N )
5033, 49breqtrd 3949 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  N )
5127, 50jca 304 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  (
( K  x.  M
)  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  N ) )
52 dvdslegcd 11642 . . . . . . . 8  |-  ( ( ( ( M  gcd  ( ( K  x.  M )  +  N
) )  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  -> 
( ( ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N
) )  ||  N
)  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  <_  ( M  gcd  N ) ) )
5352ex 114 . . . . . . 7  |-  ( ( ( M  gcd  (
( K  x.  M
)  +  N ) )  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( (
( M  gcd  (
( K  x.  M
)  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  N )  -> 
( M  gcd  (
( K  x.  M
)  +  N ) )  <_  ( M  gcd  N ) ) ) )
5430, 9, 10, 53syl3anc 1216 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( (
( M  gcd  (
( K  x.  M
)  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  N )  -> 
( M  gcd  (
( K  x.  M
)  +  N ) )  <_  ( M  gcd  N ) ) ) )
5551, 54mpid 42 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  <_  ( M  gcd  N ) ) )
5624, 55anim12d 333 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  (
( M  gcd  N
)  <_  ( M  gcd  ( ( K  x.  M )  +  N
) )  /\  ( M  gcd  ( ( K  x.  M )  +  N ) )  <_ 
( M  gcd  N
) ) ) )
577nn0red 9024 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  RR )
5829nn0red 9024 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  e.  RR )
5957, 58letri3d 7872 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  =  ( M  gcd  ( ( K  x.  M )  +  N ) )  <->  ( ( M  gcd  N )  <_ 
( M  gcd  (
( K  x.  M
)  +  N ) )  /\  ( M  gcd  ( ( K  x.  M )  +  N ) )  <_ 
( M  gcd  N
) ) ) )
6056, 59sylibrd 168 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) ) ) )
61 0zd 9059 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  e.  ZZ )
62 zdceq 9119 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
639, 61, 62syl2anc 408 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M  =  0
)
64 zdceq 9119 . . . . . . 7  |-  ( ( ( ( K  x.  M )  +  N
)  e.  ZZ  /\  0  e.  ZZ )  -> DECID  ( ( K  x.  M
)  +  N )  =  0 )
6520, 61, 64syl2anc 408 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( ( K  x.  M )  +  N
)  =  0 )
66 dcan 918 . . . . . 6  |-  (DECID  M  =  0  ->  (DECID  ( ( K  x.  M )  +  N )  =  0  -> DECID 
( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 ) ) )
6763, 65, 66sylc 62 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 ) )
68 zdceq 9119 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
6910, 61, 68syl2anc 408 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  N  =  0
)
70 dcan 918 . . . . . 6  |-  (DECID  M  =  0  ->  (DECID  N  = 
0  -> DECID  ( M  =  0  /\  N  =  0 ) ) )
7163, 69, 70sylc 62 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
72 orandc 923 . . . . 5  |-  ( (DECID  ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  /\ DECID  ( M  =  0  /\  N  =  0 ) )  ->  (
( ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  \/  ( M  =  0  /\  N  =  0 ) )  <->  -.  ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) ) ) )
7367, 71, 72syl2anc 408 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  \/  ( M  =  0  /\  N  =  0 ) )  <->  -.  ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) ) ) )
74 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  M  = 
0 )
7574oveq2d 5783 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( K  x.  M )  =  ( K  x.  0 ) )
7634mul01d 8148 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  0 )  =  0 )
7776adantr 274 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( K  x.  0 )  =  0 )
7875, 77eqtrd 2170 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( K  x.  M )  =  0 )
7978oveq1d 5782 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( K  x.  M )  +  N )  =  ( 0  +  N ) )
8047adantr 274 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( 0  +  N )  =  N )
8179, 80eqtrd 2170 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( K  x.  M )  +  N )  =  N )
8281eqeq1d 2146 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( (
( K  x.  M
)  +  N )  =  0  <->  N  = 
0 ) )
8382pm5.32da 447 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  <->  ( M  =  0  /\  N  =  0 ) ) )
84 oveq12 5776 . . . . . . . . 9  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( 0  gcd  0 ) )
8584adantl 275 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( 0  gcd  0
) )
86 oveq12 5776 . . . . . . . . . 10  |-  ( ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  =  ( 0  gcd  0 ) )
8783, 86syl6bir 163 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  /\  N  =  0 )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  =  ( 0  gcd  0 ) ) )
8887imp 123 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  =  ( 0  gcd  0
) )
8985, 88eqtr4d 2173 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) ) )
9089ex 114 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( M  gcd  ( ( K  x.  M )  +  N ) ) ) )
9183, 90sylbid 149 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  ->  ( M  gcd  N )  =  ( M  gcd  ( ( K  x.  M )  +  N ) ) ) )
9291, 90jaod 706 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  \/  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( M  gcd  ( ( K  x.  M )  +  N ) ) ) )
9373, 92sylbird 169 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) ) ) )
94 dcn 827 . . . . . 6  |-  (DECID  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  -> DECID  -.  ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 ) )
9567, 94syl 14 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 ) )
96 dcn 827 . . . . . 6  |-  (DECID  ( M  =  0  /\  N  =  0 )  -> DECID  -.  ( M  =  0  /\  N  =  0
) )
9771, 96syl 14 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  -.  ( M  =  0  /\  N  =  0 ) )
98 dcan 918 . . . . 5  |-  (DECID  -.  ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  ->  (DECID  -.  ( M  =  0  /\  N  =  0 )  -> DECID  ( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) ) ) )
9995, 97, 98sylc 62 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) ) )
100 exmiddc 821 . . . 4  |-  (DECID  ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) )  \/  -.  ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) ) ) )
10199, 100syl 14 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) )  \/  -.  ( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) ) ) )
10260, 93, 101mpjaod 707 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) ) )
10340, 14addcomd 7906 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  N )  =  ( N  +  ( K  x.  M
) ) )
104103oveq2d 5783 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  =  ( M  gcd  ( N  +  ( K  x.  M ) ) ) )
105102, 104eqtrd 2170 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  ( M  gcd  ( N  +  ( K  x.  M ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697  DECID wdc 819    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   0cc0 7613   1c1 7614    + caddc 7616    x. cmul 7618    <_ cle 7794   -ucneg 7927   NN0cn0 8970   ZZcz 9047    || cdvds 11482    gcd cgcd 11624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-gcd 11625
This theorem is referenced by:  gcdadd  11662  gcdid  11663  modgcd  11668  gcdmultipled  11670  gcdmultiple  11697
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