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Theorem mulgcddvds 12111
Description: One half of rpmulgcd2 12112, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)
Assertion
Ref Expression
mulgcddvds  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )

Proof of Theorem mulgcddvds
StepHypRef Expression
1 simp1 998 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
2 simp2 999 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
3 simp3 1000 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
42, 3zmulcld 9398 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 11986 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e. 
NN0 )
65nn0zd 9390 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e.  ZZ )
7 dvds0 11830 . . . . 5  |-  ( ( K  gcd  ( M  x.  N ) )  e.  ZZ  ->  ( K  gcd  ( M  x.  N ) )  ||  0 )
86, 7syl 14 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  0 )
98adantr 276 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  0
)
10 oveq2 5898 . . . 4  |-  ( ( K  gcd  N )  =  0  ->  (
( K  gcd  M
)  x.  ( K  gcd  N ) )  =  ( ( K  gcd  M )  x.  0 ) )
111, 2gcdcld 11986 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e. 
NN0 )
1211nn0cnd 9248 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  CC )
1312mul01d 8367 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  x.  0 )  =  0 )
1410, 13sylan9eqr 2243 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  0 )
159, 14breqtrrd 4045 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) )
166adantr 276 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  ZZ )
1716zcnd 9393 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  CC )
181, 3gcdcld 11986 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e. 
NN0 )
1918nn0zd 9390 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  ZZ )
2019adantr 276 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  ZZ )
2120zcnd 9393 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  CC )
22 0zd 9282 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  e.  ZZ )
23 zapne 9344 . . . . . 6  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( K  gcd  N ) #  0  <->  ( K  gcd  N )  =/=  0
) )
2419, 22, 23syl2anc 411 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
) #  0  <->  ( K  gcd  N )  =/=  0
) )
2524biimpar 297 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N ) #  0 )
2617, 21, 25divcanap1d 8765 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) )  =  ( K  gcd  ( M  x.  N
) ) )
27 gcddvds 11981 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( ( K  gcd  ( M  x.  N ) )  ||  K  /\  ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  N )
) )
281, 4, 27syl2anc 411 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  /\  ( K  gcd  ( M  x.  N ) ) 
||  ( M  x.  N ) ) )
2928simpld 112 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  K )
306, 1, 19, 29dvdsmultr1d 11856 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( K  x.  ( K  gcd  N ) ) )
3130adantr 276 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  ( K  x.  ( K  gcd  N ) ) )
3226, 31eqbrtrd 4039 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( K  x.  ( K  gcd  N ) ) )
33 gcddvds 11981 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  N ) )
341, 3, 33syl2anc 411 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  K  /\  ( K  gcd  N ) 
||  N ) )
3534simpld 112 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  K )
3634simprd 114 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  N )
37 dvdsmultr2 11857 . . . . . . . . . . . 12  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  N  ->  ( K  gcd  N ) 
||  ( M  x.  N ) ) )
3819, 2, 3, 37syl3anc 1248 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  N  ->  ( K  gcd  N ) 
||  ( M  x.  N ) ) )
3936, 38mpd 13 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  ( M  x.  N
) )
40 dvdsgcd 12030 . . . . . . . . . . 11  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  K  e.  ZZ  /\  ( M  x.  N )  e.  ZZ )  ->  (
( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  ( M  x.  N ) )  -> 
( K  gcd  N
)  ||  ( K  gcd  ( M  x.  N
) ) ) )
4119, 1, 4, 40syl3anc 1248 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  ( M  x.  N ) )  -> 
( K  gcd  N
)  ||  ( K  gcd  ( M  x.  N
) ) ) )
4235, 39, 41mp2and 433 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) ) )
4342adantr 276 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) ) )
44 simpr 110 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  =/=  0 )
45 dvdsval2 11814 . . . . . . . . 9  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  ( K  gcd  N )  =/=  0  /\  ( K  gcd  ( M  x.  N ) )  e.  ZZ )  ->  (
( K  gcd  N
)  ||  ( K  gcd  ( M  x.  N
) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  e.  ZZ ) )
4620, 44, 16, 45syl3anc 1248 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) )  <-> 
( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ ) )
4743, 46mpbid 147 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  e.  ZZ )
481adantr 276 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  K  e.  ZZ )
49 dvdsmulcr 11845 . . . . . . 7  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ  /\  K  e.  ZZ  /\  ( ( K  gcd  N )  e.  ZZ  /\  ( K  gcd  N )  =/=  0 ) )  -> 
( ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( K  x.  ( K  gcd  N ) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K
) )
5047, 48, 20, 44, 49syl112anc 1252 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( ( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) )  ||  ( K  x.  ( K  gcd  N ) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K
) )
5132, 50mpbid 147 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K
)
52 nn0abscl 11111 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
532, 52syl 14 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  M )  e. 
NN0 )
5453nn0zd 9390 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  M )  e.  ZZ )
55 dvdsmultr2 11857 . . . . . . . . . . . . 13  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  ( abs `  M )  e.  ZZ  /\  K  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  ->  ( K  gcd  ( M  x.  N ) ) 
||  ( ( abs `  M )  x.  K
) ) )
566, 54, 1, 55syl3anc 1248 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  ->  ( K  gcd  ( M  x.  N ) ) 
||  ( ( abs `  M )  x.  K
) ) )
5729, 56mpd 13 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  K ) )
5828simprd 114 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  N
) )
59 iddvds 11828 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  ||  M )
602, 59syl 14 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  M )
61 dvdsabsb 11834 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M  ||  M  <->  M 
||  ( abs `  M
) ) )
622, 2, 61syl2anc 411 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  M  <->  M  ||  ( abs `  M ) ) )
6360, 62mpbid 147 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( abs `  M
) )
64 dvdsmulc 11843 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  ( abs `  M )  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( abs `  M
)  ->  ( M  x.  N )  ||  (
( abs `  M
)  x.  N ) ) )
652, 54, 3, 64syl3anc 1248 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( abs `  M
)  ->  ( M  x.  N )  ||  (
( abs `  M
)  x.  N ) ) )
6663, 65mpd 13 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  ||  ( ( abs `  M
)  x.  N ) )
6754, 3zmulcld 9398 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  N )  e.  ZZ )
68 dvdstr 11852 . . . . . . . . . . . . 13  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ  /\  ( ( abs `  M
)  x.  N )  e.  ZZ )  -> 
( ( ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  N
)  /\  ( M  x.  N )  ||  (
( abs `  M
)  x.  N ) )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( abs `  M
)  x.  N ) ) )
696, 4, 67, 68syl3anc 1248 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  N )  /\  ( M  x.  N
)  ||  ( ( abs `  M )  x.  N ) )  -> 
( K  gcd  ( M  x.  N )
)  ||  ( ( abs `  M )  x.  N ) ) )
7058, 66, 69mp2and 433 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  N ) )
7154, 1zmulcld 9398 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  K )  e.  ZZ )
72 dvdsgcd 12030 . . . . . . . . . . . 12  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  ( ( abs `  M
)  x.  K )  e.  ZZ  /\  (
( abs `  M
)  x.  N )  e.  ZZ )  -> 
( ( ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  K )  /\  ( K  gcd  ( M  x.  N
) )  ||  (
( abs `  M
)  x.  N ) )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) ) ) )
736, 71, 67, 72syl3anc 1248 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( K  gcd  ( M  x.  N
) )  ||  (
( abs `  M
)  x.  K )  /\  ( K  gcd  ( M  x.  N
) )  ||  (
( abs `  M
)  x.  N ) )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) ) ) )
7457, 70, 73mp2and 433 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( ( abs `  M )  x.  K
)  gcd  ( ( abs `  M )  x.  N ) ) )
7518nn0red 9247 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  RR )
7618nn0ge0d 9249 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  <_  ( K  gcd  N
) )
7775, 76absidd 11193 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( K  gcd  N ) )  =  ( K  gcd  N ) )
7877oveq2d 5906 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  ( abs `  ( K  gcd  N
) ) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
792zcnd 9393 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
8018nn0cnd 9248 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  CC )
8179, 80absmuld 11220 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  ( K  gcd  N ) ) )  =  ( ( abs `  M
)  x.  ( abs `  ( K  gcd  N
) ) ) )
82 mulgcd 12034 . . . . . . . . . . . 12  |-  ( ( ( abs `  M
)  e.  NN0  /\  K  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
8353, 1, 3, 82syl3anc 1248 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
8478, 81, 833eqtr4d 2231 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  ( K  gcd  N ) ) )  =  ( ( ( abs `  M
)  x.  K )  gcd  ( ( abs `  M )  x.  N
) ) )
8574, 84breqtrrd 4045 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) )
862, 19zmulcld 9398 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  ( K  gcd  N ) )  e.  ZZ )
87 dvdsabsb 11834 . . . . . . . . . 10  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  ( M  x.  ( K  gcd  N ) )  e.  ZZ )  -> 
( ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  ( K  gcd  N ) )  <->  ( K  gcd  ( M  x.  N
) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) ) )
886, 86, 87syl2anc 411 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  ( M  x.  ( K  gcd  N
) )  <->  ( K  gcd  ( M  x.  N
) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) ) )
8985, 88mpbird 167 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  ( K  gcd  N ) ) )
9089adantr 276 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  ( K  gcd  N ) ) )
9126, 90eqbrtrd 4039 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( M  x.  ( K  gcd  N ) ) )
922adantr 276 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  M  e.  ZZ )
93 dvdsmulcr 11845 . . . . . . 7  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ  /\  M  e.  ZZ  /\  ( ( K  gcd  N )  e.  ZZ  /\  ( K  gcd  N )  =/=  0 ) )  -> 
( ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( M  x.  ( K  gcd  N ) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  M
) )
9447, 92, 20, 44, 93syl112anc 1252 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( ( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) )  ||  ( M  x.  ( K  gcd  N ) )  <->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  M
) )
9591, 94mpbid 147 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  M
)
96 dvdsgcd 12030 . . . . . 6  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
( ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K  /\  ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) ) 
||  M )  -> 
( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) ) 
||  ( K  gcd  M ) ) )
9747, 48, 92, 96syl3anc 1248 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( ( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) ) 
||  K  /\  (
( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) ) 
||  M )  -> 
( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) ) 
||  ( K  gcd  M ) ) )
9851, 95, 97mp2and 433 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  ( K  gcd  M ) )
9911nn0zd 9390 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  ZZ )
10099adantr 276 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  M )  e.  ZZ )
101 dvdsmulc 11843 . . . . 5  |-  ( ( ( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) )  e.  ZZ  /\  ( K  gcd  M )  e.  ZZ  /\  ( K  gcd  N )  e.  ZZ )  ->  (
( ( K  gcd  ( M  x.  N
) )  /  ( K  gcd  N ) ) 
||  ( K  gcd  M )  ->  ( (
( K  gcd  ( M  x.  N )
)  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) )  ||  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) ) )
10247, 100, 20, 101syl3anc 1248 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  ||  ( K  gcd  M )  -> 
( ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  x.  ( K  gcd  N ) ) 
||  ( ( K  gcd  M )  x.  ( K  gcd  N
) ) ) )
10398, 102mpd 13 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( ( K  gcd  ( M  x.  N ) )  /  ( K  gcd  N ) )  x.  ( K  gcd  N ) ) 
||  ( ( K  gcd  M )  x.  ( K  gcd  N
) ) )
10426, 103eqbrtrrd 4041 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  (
( K  gcd  M
)  x.  ( K  gcd  N ) ) )
105 zdceq 9345 . . . 4  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  0  e.  ZZ )  -> DECID  ( K  gcd  N )  =  0 )
10619, 22, 105syl2anc 411 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( K  gcd  N
)  =  0 )
107 exmiddc 837 . . . 4  |-  (DECID  ( K  gcd  N )  =  0  ->  ( ( K  gcd  N )  =  0  \/  -.  ( K  gcd  N )  =  0 ) )
108 df-ne 2360 . . . . 5  |-  ( ( K  gcd  N )  =/=  0  <->  -.  ( K  gcd  N )  =  0 )
109108orbi2i 763 . . . 4  |-  ( ( ( K  gcd  N
)  =  0  \/  ( K  gcd  N
)  =/=  0 )  <-> 
( ( K  gcd  N )  =  0  \/ 
-.  ( K  gcd  N )  =  0 ) )
110107, 109sylibr 134 . . 3  |-  (DECID  ( K  gcd  N )  =  0  ->  ( ( K  gcd  N )  =  0  \/  ( K  gcd  N )  =/=  0 ) )
111106, 110syl 14 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  =  0  \/  ( K  gcd  N
)  =/=  0 ) )
11215, 104, 111mpjaodan 799 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( K  gcd  M )  x.  ( K  gcd  N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835    /\ w3a 979    = wceq 1363    e. wcel 2159    =/= wne 2359   class class class wbr 4017   ` cfv 5230  (class class class)co 5890   0cc0 7828    x. cmul 7833   # cap 8555    / cdiv 8646   NN0cn0 9193   ZZcz 9270   abscabs 11023    || cdvds 11811    gcd cgcd 11960
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-sup 7000  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-q 9637  df-rp 9671  df-fz 10026  df-fzo 10160  df-fl 10287  df-mod 10340  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-dvds 11812  df-gcd 11961
This theorem is referenced by:  rpmulgcd2  12112  rpmul  12115
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