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Theorem mulgcddvds 11702
Description: One half of rpmulgcd2 11703, 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 966 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
2 simp2 967 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
3 simp3 968 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
42, 3zmulcld 9147 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 11584 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e. 
NN0 )
65nn0zd 9139 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e.  ZZ )
7 dvds0 11435 . . . . 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 5750 . . . 4  |-  ( ( K  gcd  N )  =  0  ->  (
( K  gcd  M
)  x.  ( K  gcd  N ) )  =  ( ( K  gcd  M )  x.  0 ) )
111, 2gcdcld 11584 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e. 
NN0 )
1211nn0cnd 9000 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  CC )
1312mul01d 8123 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  x.  0 )  =  0 )
1410, 13sylan9eqr 2172 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  0 )
159, 14breqtrrd 3926 . 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 9142 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  CC )
181, 3gcdcld 11584 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e. 
NN0 )
1918nn0zd 9139 . . . . . 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 9142 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  CC )
22 0zd 9034 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  e.  ZZ )
23 zapne 9093 . . . . . 6  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( K  gcd  N ) #  0  <->  ( K  gcd  N )  =/=  0
) )
2419, 22, 23syl2anc 408 . . . . 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 8519 . . 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 11579 . . . . . . . . . . 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 408 . . . . . . . . . 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 11459 . . . . . . . 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 3920 . . . . . 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 11579 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  N ) )
341, 3, 33syl2anc 408 . . . . . . . . . . 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 11460 . . . . . . . . . . . 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 1201 . . . . . . . . . . 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 11627 . . . . . . . . . . 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 1201 . . . . . . . . . 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 429 . . . . . . . . 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 11423 . . . . . . . . 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 1201 . . . . . . . 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 11450 . . . . . . 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 1205 . . . . . 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 10825 . . . . . . . . . . . . . . 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 9139 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  M )  e.  ZZ )
55 dvdsmultr2 11460 . . . . . . . . . . . . 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 1201 . . . . . . . . . . . 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 11433 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  ||  M )
602, 59syl 14 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  M )
61 dvdsabsb 11439 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M  ||  M  <->  M 
||  ( abs `  M
) ) )
622, 2, 61syl2anc 408 . . . . . . . . . . . . . 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 11448 . . . . . . . . . . . . . 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 1201 . . . . . . . . . . . . 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 9147 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  N )  e.  ZZ )
68 dvdstr 11457 . . . . . . . . . . . . 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 1201 . . . . . . . . . . . 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 429 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  N ) )
7154, 1zmulcld 9147 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  K )  e.  ZZ )
72 dvdsgcd 11627 . . . . . . . . . . . 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 1201 . . . . . . . . . . 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 429 . . . . . . . . . 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 8999 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  RR )
7618nn0ge0d 9001 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  <_  ( K  gcd  N
) )
7775, 76absidd 10907 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( K  gcd  N ) )  =  ( K  gcd  N ) )
7877oveq2d 5758 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  ( abs `  ( K  gcd  N
) ) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
792zcnd 9142 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
8018nn0cnd 9000 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  CC )
8179, 80absmuld 10934 . . . . . . . . . . 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 11631 . . . . . . . . . . . 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 1201 . . . . . . . . . . 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 2160 . . . . . . . . . 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 3926 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) )
862, 19zmulcld 9147 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  ( K  gcd  N ) )  e.  ZZ )
87 dvdsabsb 11439 . . . . . . . . . 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 408 . . . . . . . . 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 3920 . . . . . 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 11450 . . . . . . 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 1205 . . . . . 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 11627 . . . . . 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 1201 . . . . 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 429 . . . 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 9139 . . . . . 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 11448 . . . . 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 1201 . . . 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 3922 . 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 9094 . . . 4  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  0  e.  ZZ )  -> DECID  ( K  gcd  N )  =  0 )
10619, 22, 105syl2anc 408 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( K  gcd  N
)  =  0 )
107 exmiddc 806 . . . 4  |-  (DECID  ( K  gcd  N )  =  0  ->  ( ( K  gcd  N )  =  0  \/  -.  ( K  gcd  N )  =  0 ) )
108 df-ne 2286 . . . . 5  |-  ( ( K  gcd  N )  =/=  0  <->  -.  ( K  gcd  N )  =  0 )
109108orbi2i 736 . . . 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 772 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 682  DECID wdc 804    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   0cc0 7588    x. cmul 7593   # cap 8311    / cdiv 8400   NN0cn0 8945   ZZcz 9022   abscabs 10737    || cdvds 11420    gcd cgcd 11562
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-gcd 11563
This theorem is referenced by:  rpmulgcd2  11703  rpmul  11706
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