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Theorem mulgcddvds 12035
Description: One half of rpmulgcd2 12036, 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 992 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
2 simp2 993 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
3 simp3 994 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
42, 3zmulcld 9327 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 11910 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e. 
NN0 )
65nn0zd 9319 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e.  ZZ )
7 dvds0 11755 . . . . 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 274 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  0
)
10 oveq2 5858 . . . 4  |-  ( ( K  gcd  N )  =  0  ->  (
( K  gcd  M
)  x.  ( K  gcd  N ) )  =  ( ( K  gcd  M )  x.  0 ) )
111, 2gcdcld 11910 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e. 
NN0 )
1211nn0cnd 9177 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  CC )
1312mul01d 8299 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  x.  0 )  =  0 )
1410, 13sylan9eqr 2225 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  0 )
159, 14breqtrrd 4015 . 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 274 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  ZZ )
1716zcnd 9322 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  CC )
181, 3gcdcld 11910 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e. 
NN0 )
1918nn0zd 9319 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  ZZ )
2019adantr 274 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  ZZ )
2120zcnd 9322 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  CC )
22 0zd 9211 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  e.  ZZ )
23 zapne 9273 . . . . . 6  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( K  gcd  N ) #  0  <->  ( K  gcd  N )  =/=  0
) )
2419, 22, 23syl2anc 409 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
) #  0  <->  ( K  gcd  N )  =/=  0
) )
2524biimpar 295 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N ) #  0 )
2617, 21, 25divcanap1d 8695 . . 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 11905 . . . . . . . . . . 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 409 . . . . . . . . . 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 111 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  K )
306, 1, 19, 29dvdsmultr1d 11781 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( K  x.  ( K  gcd  N ) ) )
3130adantr 274 . . . . . . 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 4009 . . . . . 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 11905 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  N ) )
341, 3, 33syl2anc 409 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  K  /\  ( K  gcd  N ) 
||  N ) )
3534simpld 111 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  K )
3634simprd 113 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  N )
37 dvdsmultr2 11782 . . . . . . . . . . . 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 1233 . . . . . . . . . . 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 11954 . . . . . . . . . . 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 1233 . . . . . . . . . 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 431 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) ) )
4342adantr 274 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) ) )
44 simpr 109 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  =/=  0 )
45 dvdsval2 11739 . . . . . . . . 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 1233 . . . . . . . 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 146 . . . . . . 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 274 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  K  e.  ZZ )
49 dvdsmulcr 11770 . . . . . . 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 1237 . . . . . 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 146 . . . . 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 11036 . . . . . . . . . . . . . . 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 9319 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  M )  e.  ZZ )
55 dvdsmultr2 11782 . . . . . . . . . . . . 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 1233 . . . . . . . . . . . 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 113 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  N
) )
59 iddvds 11753 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  ||  M )
602, 59syl 14 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  M )
61 dvdsabsb 11759 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M  ||  M  <->  M 
||  ( abs `  M
) ) )
622, 2, 61syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  M  <->  M  ||  ( abs `  M ) ) )
6360, 62mpbid 146 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( abs `  M
) )
64 dvdsmulc 11768 . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . 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 9327 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  N )  e.  ZZ )
68 dvdstr 11777 . . . . . . . . . . . . 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 1233 . . . . . . . . . . . 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 431 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  N ) )
7154, 1zmulcld 9327 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  K )  e.  ZZ )
72 dvdsgcd 11954 . . . . . . . . . . . 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 1233 . . . . . . . . . . 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 431 . . . . . . . . . 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 9176 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  RR )
7618nn0ge0d 9178 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  <_  ( K  gcd  N
) )
7775, 76absidd 11118 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( K  gcd  N ) )  =  ( K  gcd  N ) )
7877oveq2d 5866 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  ( abs `  ( K  gcd  N
) ) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
792zcnd 9322 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
8018nn0cnd 9177 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  CC )
8179, 80absmuld 11145 . . . . . . . . . . 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 11958 . . . . . . . . . . . 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 1233 . . . . . . . . . . 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 2213 . . . . . . . . . 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 4015 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) )
862, 19zmulcld 9327 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  ( K  gcd  N ) )  e.  ZZ )
87 dvdsabsb 11759 . . . . . . . . . 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 409 . . . . . . . . 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 166 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  ( K  gcd  N ) ) )
9089adantr 274 . . . . . . 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 4009 . . . . . 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 274 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  M  e.  ZZ )
93 dvdsmulcr 11770 . . . . . . 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 1237 . . . . . 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 146 . . . . 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 11954 . . . . . 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 1233 . . . . 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 431 . . . 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 9319 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  ZZ )
10099adantr 274 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  M )  e.  ZZ )
101 dvdsmulc 11768 . . . . 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 1233 . . . 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 4011 . 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 9274 . . . 4  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  0  e.  ZZ )  -> DECID  ( K  gcd  N )  =  0 )
10619, 22, 105syl2anc 409 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( K  gcd  N
)  =  0 )
107 exmiddc 831 . . . 4  |-  (DECID  ( K  gcd  N )  =  0  ->  ( ( K  gcd  N )  =  0  \/  -.  ( K  gcd  N )  =  0 ) )
108 df-ne 2341 . . . . 5  |-  ( ( K  gcd  N )  =/=  0  <->  -.  ( K  gcd  N )  =  0 )
109108orbi2i 757 . . . 4  |-  ( ( ( K  gcd  N
)  =  0  \/  ( K  gcd  N
)  =/=  0 )  <-> 
( ( K  gcd  N )  =  0  \/ 
-.  ( K  gcd  N )  =  0 ) )
110107, 109sylibr 133 . . 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 793 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 103    <-> wb 104    \/ wo 703  DECID wdc 829    /\ w3a 973    = wceq 1348    e. wcel 2141    =/= wne 2340   class class class wbr 3987   ` cfv 5196  (class class class)co 5850   0cc0 7761    x. cmul 7766   # cap 8487    / cdiv 8576   NN0cn0 9122   ZZcz 9199   abscabs 10948    || cdvds 11736    gcd cgcd 11884
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-sup 6957  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-fl 10213  df-mod 10266  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-dvds 11737  df-gcd 11885
This theorem is referenced by:  rpmulgcd2  12036  rpmul  12039
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