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Theorem gcdaddm 12020
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 11999 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
213adant1 1017 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  N ) )
32simpld 112 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  M )
4 simp1 999 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
5 1zzd 9311 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  1  e.  ZZ )
6 gcdcl 12002 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
763adant1 1017 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e. 
NN0 )
87nn0zd 9404 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  ZZ )
9 simp2 1000 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
10 simp3 1001 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
11 dvds2ln 11866 . . . . . . . . . 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 1256 . . . . . . . . 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 9407 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
1514mulid2d 8007 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
1  x.  N )  =  N )
1615oveq2d 5913 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  ( 1  x.  N ) )  =  ( ( K  x.  M )  +  N ) )
1713, 16breqtrd 4044 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  ( ( K  x.  M )  +  N
) )
183, 17jca 306 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  ( ( K  x.  M )  +  N ) ) )
194, 9zmulcld 9412 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  ZZ )
2019, 10zaddcld 9410 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  N )  e.  ZZ )
21 dvdslegcd 12000 . . . . . . . 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 115 . . . . . . 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 1249 . . . . . 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 11999 . . . . . . . . 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 411 . . . . . . . 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 112 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  M )
284znegcld 9408 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  -u K  e.  ZZ )
299, 20gcdcld 12004 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  e. 
NN0 )
3029nn0zd 9404 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  e.  ZZ )
31 dvds2ln 11866 . . . . . . . . . 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 1256 . . . . . . . . 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 9407 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  CC )
359zcnd 9407 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
3634, 35mulneg1d 8399 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u K  x.  M )  =  -u ( K  x.  M ) )
3720zcnd 9407 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  N )  e.  CC )
3837mulid2d 8007 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
1  x.  ( ( K  x.  M )  +  N ) )  =  ( ( K  x.  M )  +  N ) )
3936, 38oveq12d 5915 . . . . . . . . 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 8009 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  M )  e.  CC )
4140negcld 8286 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  -u ( K  x.  M )  e.  CC )
4240, 41addcomd 8139 . . . . . . . . . . . 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 8289 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  -u ( K  x.  M )
)  =  0 )
4442, 43eqtr3d 2224 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u ( K  x.  M
)  +  ( K  x.  M ) )  =  0 )
4544oveq1d 5912 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -u ( K  x.  M )  +  ( K  x.  M ) )  +  N )  =  ( 0  +  N ) )
4641, 40, 14addassd 8011 . . . . . . . . . 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 ) ) )
4714addlidd 8138 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
0  +  N )  =  N )
4845, 46, 473eqtr3d 2230 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u ( K  x.  M
)  +  ( ( K  x.  M )  +  N ) )  =  N )
4939, 48eqtrd 2222 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( -u K  x.  M
)  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) )  =  N )
5033, 49breqtrd 4044 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  N )
5127, 50jca 306 . . . . . 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 12000 . . . . . . . 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 115 . . . . . . 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, 53syld3an1 1295 . . . . . 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 335 . . . 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 9261 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  RR )
5829nn0red 9261 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  e.  RR )
5957, 58letri3d 8104 . . . 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 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 ) ) ) )
61 0zd 9296 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  e.  ZZ )
62 zdceq 9359 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
639, 61, 62syl2anc 411 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M  =  0
)
64 zdceq 9359 . . . . . . 7  |-  ( ( ( ( K  x.  M )  +  N
)  e.  ZZ  /\  0  e.  ZZ )  -> DECID  ( ( K  x.  M
)  +  N )  =  0 )
6520, 61, 64syl2anc 411 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( ( K  x.  M )  +  N
)  =  0 )
6663, 65dcand 934 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 ) )
67 zdceq 9359 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
6810, 61, 67syl2anc 411 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  N  =  0
)
6963, 68dcand 934 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
70 orandc 941 . . . . 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 ) ) ) )
7166, 69, 70syl2anc 411 . . . 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 ) ) ) )
72 simpr 110 . . . . . . . . . . . 12  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  M  = 
0 )
7372oveq2d 5913 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( K  x.  M )  =  ( K  x.  0 ) )
7434mul01d 8381 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  0 )  =  0 )
7574adantr 276 . . . . . . . . . . 11  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( K  x.  0 )  =  0 )
7673, 75eqtrd 2222 . . . . . . . . . 10  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( K  x.  M )  =  0 )
7776oveq1d 5912 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( K  x.  M )  +  N )  =  ( 0  +  N ) )
7847adantr 276 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( 0  +  N )  =  N )
7977, 78eqtrd 2222 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( K  x.  M )  +  N )  =  N )
8079eqeq1d 2198 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( (
( K  x.  M
)  +  N )  =  0  <->  N  = 
0 ) )
8180pm5.32da 452 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  <->  ( M  =  0  /\  N  =  0 ) ) )
82 oveq12 5906 . . . . . . . . 9  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( 0  gcd  0 ) )
8382adantl 277 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( 0  gcd  0
) )
84 oveq12 5906 . . . . . . . . . 10  |-  ( ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  =  ( 0  gcd  0 ) )
8581, 84biimtrrdi 164 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  /\  N  =  0 )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  =  ( 0  gcd  0 ) ) )
8685imp 124 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  =  ( 0  gcd  0
) )
8783, 86eqtr4d 2225 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) ) )
8887ex 115 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( M  gcd  ( ( K  x.  M )  +  N ) ) ) )
8981, 88sylbid 150 . . . . 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 ) ) ) )
9089, 88jaod 718 . . . 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 ) ) ) )
9171, 90sylbird 170 . . 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 ) ) ) )
92 dcn 843 . . . . . 6  |-  (DECID  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  -> DECID  -.  ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 ) )
9366, 92syl 14 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 ) )
94 dcn 843 . . . . . 6  |-  (DECID  ( M  =  0  /\  N  =  0 )  -> DECID  -.  ( M  =  0  /\  N  =  0
) )
9569, 94syl 14 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  -.  ( M  =  0  /\  N  =  0 ) )
9693, 95dcand 934 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) ) )
97 exmiddc 837 . . . 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 ) ) ) )
9896, 97syl 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 ) ) ) )
9960, 91, 98mpjaod 719 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) ) )
10040, 14addcomd 8139 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  +  N )  =  ( N  +  ( K  x.  M
) ) )
101100oveq2d 5913 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  =  ( M  gcd  ( N  +  ( K  x.  M ) ) ) )
10299, 101eqtrd 2222 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 104    <-> wb 105    \/ wo 709  DECID wdc 835    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5897   0cc0 7842   1c1 7843    + caddc 7845    x. cmul 7847    <_ cle 8024   -ucneg 8160   NN0cn0 9207   ZZcz 9284    || cdvds 11829    gcd cgcd 11978
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-sup 7014  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-fl 10303  df-mod 10356  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-dvds 11830  df-gcd 11979
This theorem is referenced by:  gcdadd  12021  gcdid  12022  modgcd  12027  gcdmultipled  12029  gcdmultiple  12056  pythagtriplem4  12303
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