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Theorem absmulgcd 13730
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 13700 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
2 nn0re 10587 . . . . . 6  |-  ( ( M  gcd  N )  e.  NN0  ->  ( M  gcd  N )  e.  RR )
3 nn0ge0 10604 . . . . . 6  |-  ( ( M  gcd  N )  e.  NN0  ->  0  <_ 
( M  gcd  N
) )
42, 3absidd 12908 . . . . 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 6106 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  K
)  x.  ( abs `  ( M  gcd  N
) ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
763adant1 1006 . 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 10650 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  CC )
91nn0cnd 10637 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
10 absmul 12782 . . . 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 1183 . 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 10650 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 10650 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 absmul 12782 . . . . . . 7  |-  ( ( K  e.  CC  /\  M  e.  CC )  ->  ( abs `  ( K  x.  M )
)  =  ( ( abs `  K )  x.  ( abs `  M
) ) )
16 absmul 12782 . . . . . . 7  |-  ( ( K  e.  CC  /\  N  e.  CC )  ->  ( abs `  ( K  x.  N )
)  =  ( ( abs `  K )  x.  ( abs `  N
) ) )
1715, 16oveqan12d 6109 . . . . . 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 1273 . . . . 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 1260 . . . 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 10692 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
21 zmulcl 10692 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
22 gcdabs 13716 . . . . . 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 1273 . . . 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 12800 . . . . 5  |-  ( K  e.  ZZ  ->  ( abs `  K )  e. 
NN0 )
26 nn0abscl 12800 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
2726nn0zd 10744 . . . . 5  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
28 nn0abscl 12800 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
2928nn0zd 10744 . . . . 5  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
30 mulgcd 13729 . . . . 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 1260 . . . 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 2482 . . 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 13716 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
34333adant1 1006 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
3534oveq2d 6106 . . 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 2474 . 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 2485 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 965    = wceq 1369    e. wcel 1756   ` cfv 5417  (class class class)co 6090   CCcc 9279    x. cmul 9286   NN0cn0 10578   ZZcz 10645   abscabs 12722    gcd cgcd 13689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-dvds 13535  df-gcd 13690
This theorem is referenced by:  coprmdvds  13787
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