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Theorem mulgcddvds 12783
Description: One half of rpmulgcd2 12784, 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 955 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  K  e.  ZZ )
2 simp2 956 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
3 simp3 957 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
42, 3zmulcld 10123 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
51, 4gcdcld 12697 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e. 
NN0 )
65nn0zd 10115 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  e.  ZZ )
7 dvds0 12544 . . . . 5  |-  ( ( K  gcd  ( M  x.  N ) )  e.  ZZ  ->  ( K  gcd  ( M  x.  N ) )  ||  0 )
86, 7syl 15 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  0 )
98adantr 451 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  0
)
10 oveq2 5866 . . . 4  |-  ( ( K  gcd  N )  =  0  ->  (
( K  gcd  M
)  x.  ( K  gcd  N ) )  =  ( ( K  gcd  M )  x.  0 ) )
111, 2gcdcld 12697 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e. 
NN0 )
1211nn0cnd 10020 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  CC )
1312mul01d 9011 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  M
)  x.  0 )  =  0 )
1410, 13sylan9eqr 2337 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =  0 )  ->  ( ( K  gcd  M )  x.  ( K  gcd  N
) )  =  0 )
159, 14breqtrrd 4049 . 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 451 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  ZZ )
1716zcnd 10118 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  e.  CC )
181, 3gcdcld 12697 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e. 
NN0 )
1918nn0zd 10115 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  ZZ )
2019adantr 451 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  ZZ )
2120zcnd 10118 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  e.  CC )
22 simpr 447 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  =/=  0 )
2317, 21, 22divcan1d 9537 . . 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
) ) )
24 gcddvds 12694 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( ( K  gcd  ( M  x.  N ) )  ||  K  /\  ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  N )
) )
251, 4, 24syl2anc 642 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  /\  ( K  gcd  ( M  x.  N ) ) 
||  ( M  x.  N ) ) )
2625simpld 445 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  K )
27 dvdsmultr1 12563 . . . . . . . . . 10  |-  ( ( ( K  gcd  ( M  x.  N )
)  e.  ZZ  /\  K  e.  ZZ  /\  ( K  gcd  N )  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  ->  ( K  gcd  ( M  x.  N ) ) 
||  ( K  x.  ( K  gcd  N ) ) ) )
286, 1, 19, 27syl3anc 1182 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  ( M  x.  N )
)  ||  K  ->  ( K  gcd  ( M  x.  N ) ) 
||  ( K  x.  ( K  gcd  N ) ) ) )
2926, 28mpd 14 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( K  x.  ( K  gcd  N ) ) )
3029adantr 451 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  ( K  x.  ( K  gcd  N ) ) )
3123, 30eqbrtrd 4043 . . . . . 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 ) ) )
32 gcddvds 12694 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  gcd  N )  ||  K  /\  ( K  gcd  N ) 
||  N ) )
331, 3, 32syl2anc 642 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  K  /\  ( K  gcd  N ) 
||  N ) )
3433simpld 445 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  K )
3533simprd 449 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  N )
36 dvdsmultr2 12564 . . . . . . . . . . . 12  |-  ( ( ( K  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  N  ->  ( K  gcd  N ) 
||  ( M  x.  N ) ) )
3719, 2, 3, 36syl3anc 1182 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  gcd  N
)  ||  N  ->  ( K  gcd  N ) 
||  ( M  x.  N ) ) )
3835, 37mpd 14 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  ( M  x.  N
) )
39 dvdsgcd 12722 . . . . . . . . . . 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
) ) ) )
4019, 1, 4, 39syl3anc 1182 . . . . . . . . . 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
) ) ) )
4134, 38, 40mp2and 660 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) ) )
4241adantr 451 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  N )  ||  ( K  gcd  ( M  x.  N ) ) )
43 dvdsval2 12534 . . . . . . . . 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 ) )
4420, 22, 16, 43syl3anc 1182 . . . . . . . 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 ) )
4542, 44mpbid 201 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  e.  ZZ )
461adantr 451 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  K  e.  ZZ )
47 dvdsmulcr 12558 . . . . . . 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
) )
4845, 46, 20, 22, 47syl112anc 1186 . . . . . 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
) )
4931, 48mpbid 201 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  K
)
50 nn0abscl 11797 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
512, 50syl 15 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  M )  e. 
NN0 )
5251nn0zd 10115 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  M )  e.  ZZ )
53 dvdsmultr2 12564 . . . . . . . . . . . . 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
) ) )
546, 52, 1, 53syl3anc 1182 . . . . . . . . . . . 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
) ) )
5526, 54mpd 14 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  K ) )
5625simprd 449 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  N
) )
57 iddvds 12542 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  ||  M )
582, 57syl 15 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  M )
59 dvdsabsb 12548 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M  ||  M  <->  M 
||  ( abs `  M
) ) )
602, 2, 59syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  M  <->  M  ||  ( abs `  M ) ) )
6158, 60mpbid 201 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( abs `  M
) )
62 dvdsmulc 12556 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  ( abs `  M )  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( abs `  M
)  ->  ( M  x.  N )  ||  (
( abs `  M
)  x.  N ) ) )
632, 52, 3, 62syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( abs `  M
)  ->  ( M  x.  N )  ||  (
( abs `  M
)  x.  N ) ) )
6461, 63mpd 14 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  ||  ( ( abs `  M
)  x.  N ) )
6552, 3zmulcld 10123 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  N )  e.  ZZ )
66 dvdstr 12562 . . . . . . . . . . . . 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 ) ) )
676, 4, 65, 66syl3anc 1182 . . . . . . . . . . . 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 ) ) )
6856, 64, 67mp2and 660 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( abs `  M
)  x.  N ) )
6952, 1zmulcld 10123 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  K )  e.  ZZ )
70 dvdsgcd 12722 . . . . . . . . . . . 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
) ) ) )
716, 69, 65, 70syl3anc 1182 . . . . . . . . . . 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
) ) ) )
7255, 68, 71mp2and 660 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( ( ( abs `  M )  x.  K
)  gcd  ( ( abs `  M )  x.  N ) ) )
7318nn0red 10019 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  RR )
7418nn0ge0d 10021 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  0  <_  ( K  gcd  N
) )
7573, 74absidd 11905 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( K  gcd  N ) )  =  ( K  gcd  N ) )
7675oveq2d 5874 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  x.  ( abs `  ( K  gcd  N
) ) )  =  ( ( abs `  M
)  x.  ( K  gcd  N ) ) )
772zcnd 10118 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7818nn0cnd 10020 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  N )  e.  CC )
7977, 78absmuld 11936 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( M  x.  ( K  gcd  N ) ) )  =  ( ( abs `  M
)  x.  ( abs `  ( K  gcd  N
) ) ) )
80 mulgcd 12725 . . . . . . . . . . . 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 ) ) )
8151, 1, 3, 80syl3anc 1182 . . . . . . . . . . 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 ) ) )
8276, 79, 813eqtr4d 2325 . . . . . . . . . 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
) ) )
8372, 82breqtrrd 4049 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( abs `  ( M  x.  ( K  gcd  N ) ) ) )
842, 19zmulcld 10123 . . . . . . . . . 10  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  ( K  gcd  N ) )  e.  ZZ )
85 dvdsabsb 12548 . . . . . . . . . 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 ) ) ) ) )
866, 84, 85syl2anc 642 . . . . . . . . 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 ) ) ) ) )
8783, 86mpbird 223 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) )  ||  ( M  x.  ( K  gcd  N ) ) )
8887adantr 451 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  ( M  x.  N
) )  ||  ( M  x.  ( K  gcd  N ) ) )
8923, 88eqbrtrd 4043 . . . . . 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 ) ) )
902adantr 451 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  M  e.  ZZ )
91 dvdsmulcr 12558 . . . . . . 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
) )
9245, 90, 20, 22, 91syl112anc 1186 . . . . . 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
) )
9389, 92mpbid 201 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( ( K  gcd  ( M  x.  N ) )  / 
( K  gcd  N
) )  ||  M
)
94 dvdsgcd 12722 . . . . . 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 ) ) )
9545, 46, 90, 94syl3anc 1182 . . . . 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 ) ) )
9649, 93, 95mp2and 660 . . . 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 ) )
9711nn0zd 10115 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  M )  e.  ZZ )
9897adantr 451 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  gcd  N
)  =/=  0 )  ->  ( K  gcd  M )  e.  ZZ )
99 dvdsmulc 12556 . . . . 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 ) ) ) )
10045, 98, 20, 99syl3anc 1182 . . . 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
) ) ) )
10196, 100mpd 14 . . 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
) ) )
10223, 101eqbrtrrd 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 ) ) )
10315, 102pm2.61dane 2524 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:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0cc0 8737    x. cmul 8742    / cdiv 9423   NN0cn0 9965   ZZcz 10024   abscabs 11719    || cdivides 12531    gcd cgcd 12685
This theorem is referenced by:  rpmulgcd2  12784  rpmul  12802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
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