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Theorem absmulgcd 14061
Description: Distribute absolute value of multiplication over gcd. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
absmulgcd  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( abs `  ( K  x.  ( M  gcd  N ) ) ) )

Proof of Theorem absmulgcd
StepHypRef Expression
1 gcdcl 14031 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
2 nn0re 10816 . . . . . 6  |-  ( ( M  gcd  N )  e.  NN0  ->  ( M  gcd  N )  e.  RR )
3 nn0ge0 10833 . . . . . 6  |-  ( ( M  gcd  N )  e.  NN0  ->  0  <_ 
( M  gcd  N
) )
42, 3absidd 13234 . . . . 5  |-  ( ( M  gcd  N )  e.  NN0  ->  ( abs `  ( M  gcd  N
) )  =  ( M  gcd  N ) )
51, 4syl 16 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( M  gcd  N ) )  =  ( M  gcd  N ) )
65oveq2d 6311 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  K
)  x.  ( abs `  ( M  gcd  N
) ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
763adant1 1014 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  K
)  x.  ( abs `  ( M  gcd  N
) ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
8 zcn 10881 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  CC )
91nn0cnd 10866 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
10 absmul 13107 . . . 4  |-  ( ( K  e.  CC  /\  ( M  gcd  N )  e.  CC )  -> 
( abs `  ( K  x.  ( M  gcd  N ) ) )  =  ( ( abs `  K )  x.  ( abs `  ( M  gcd  N ) ) ) )
118, 9, 10syl2an 477 . . 3  |-  ( ( K  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  ( K  x.  ( M  gcd  N ) ) )  =  ( ( abs `  K )  x.  ( abs `  ( M  gcd  N ) ) ) )
12113impb 1192 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( K  x.  ( M  gcd  N ) ) )  =  ( ( abs `  K
)  x.  ( abs `  ( M  gcd  N
) ) ) )
13 zcn 10881 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 10881 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 absmul 13107 . . . . . . 7  |-  ( ( K  e.  CC  /\  M  e.  CC )  ->  ( abs `  ( K  x.  M )
)  =  ( ( abs `  K )  x.  ( abs `  M
) ) )
16 absmul 13107 . . . . . . 7  |-  ( ( K  e.  CC  /\  N  e.  CC )  ->  ( abs `  ( K  x.  N )
)  =  ( ( abs `  K )  x.  ( abs `  N
) ) )
1715, 16oveqan12d 6314 . . . . . 6  |-  ( ( ( K  e.  CC  /\  M  e.  CC )  /\  ( K  e.  CC  /\  N  e.  CC ) )  -> 
( ( abs `  ( K  x.  M )
)  gcd  ( abs `  ( K  x.  N
) ) )  =  ( ( ( abs `  K )  x.  ( abs `  M ) )  gcd  ( ( abs `  K )  x.  ( abs `  N ) ) ) )
18173impdi 1283 . . . . 5  |-  ( ( K  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( abs `  ( K  x.  M )
)  gcd  ( abs `  ( K  x.  N
) ) )  =  ( ( ( abs `  K )  x.  ( abs `  M ) )  gcd  ( ( abs `  K )  x.  ( abs `  N ) ) ) )
198, 13, 14, 18syl3an 1270 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  ( K  x.  M )
)  gcd  ( abs `  ( K  x.  N
) ) )  =  ( ( ( abs `  K )  x.  ( abs `  M ) )  gcd  ( ( abs `  K )  x.  ( abs `  N ) ) ) )
20 zmulcl 10923 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
21 zmulcl 10923 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
22 gcdabs 14047 . . . . . 6  |-  ( ( ( K  x.  M
)  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ )  ->  ( ( abs `  ( K  x.  M
) )  gcd  ( abs `  ( K  x.  N ) ) )  =  ( ( K  x.  M )  gcd  ( K  x.  N
) ) )
2320, 21, 22syl2an 477 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  ( K  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( abs `  ( K  x.  M )
)  gcd  ( abs `  ( K  x.  N
) ) )  =  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
24233impdi 1283 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  ( K  x.  M )
)  gcd  ( abs `  ( K  x.  N
) ) )  =  ( ( K  x.  M )  gcd  ( K  x.  N )
) )
25 nn0abscl 13125 . . . . 5  |-  ( K  e.  ZZ  ->  ( abs `  K )  e. 
NN0 )
26 nn0abscl 13125 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
2726nn0zd 10976 . . . . 5  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
28 nn0abscl 13125 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
2928nn0zd 10976 . . . . 5  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
30 mulgcd 14060 . . . . 5  |-  ( ( ( abs `  K
)  e.  NN0  /\  ( abs `  M )  e.  ZZ  /\  ( abs `  N )  e.  ZZ )  ->  (
( ( abs `  K
)  x.  ( abs `  M ) )  gcd  ( ( abs `  K
)  x.  ( abs `  N ) ) )  =  ( ( abs `  K )  x.  (
( abs `  M
)  gcd  ( abs `  N ) ) ) )
3125, 27, 29, 30syl3an 1270 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( abs `  K
)  x.  ( abs `  M ) )  gcd  ( ( abs `  K
)  x.  ( abs `  N ) ) )  =  ( ( abs `  K )  x.  (
( abs `  M
)  gcd  ( abs `  N ) ) ) )
3219, 24, 313eqtr3d 2516 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( ( abs `  K
)  x.  ( ( abs `  M )  gcd  ( abs `  N
) ) ) )
33 gcdabs 14047 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
34333adant1 1014 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
3534oveq2d 6311 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  K
)  x.  ( ( abs `  M )  gcd  ( abs `  N
) ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
3632, 35eqtrd 2508 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
377, 12, 363eqtr4rd 2519 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( abs `  ( K  x.  ( M  gcd  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   CCcc 9502    x. cmul 9509   NN0cn0 10807   ZZcz 10876   abscabs 13047    gcd cgcd 14020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021
This theorem is referenced by:  coprmdvds  14119
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