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Theorem absmulgcd 12726
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 12696 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
2 nn0re 9974 . . . . . 6  |-  ( ( M  gcd  N )  e.  NN0  ->  ( M  gcd  N )  e.  RR )
3 nn0ge0 9991 . . . . . 6  |-  ( ( M  gcd  N )  e.  NN0  ->  0  <_ 
( M  gcd  N
) )
42, 3absidd 11905 . . . . 5  |-  ( ( M  gcd  N )  e.  NN0  ->  ( abs `  ( M  gcd  N
) )  =  ( M  gcd  N ) )
51, 4syl 15 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( M  gcd  N ) )  =  ( M  gcd  N ) )
65oveq2d 5874 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  K
)  x.  ( abs `  ( M  gcd  N
) ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
763adant1 973 . 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 10029 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  CC )
91nn0cnd 10020 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
10 absmul 11779 . . . 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 463 . . 3  |-  ( ( K  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  ( K  x.  ( M  gcd  N ) ) )  =  ( ( abs `  K )  x.  ( abs `  ( M  gcd  N ) ) ) )
12113impb 1147 . 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 10029 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 10029 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 absmul 11779 . . . . . . 7  |-  ( ( K  e.  CC  /\  M  e.  CC )  ->  ( abs `  ( K  x.  M )
)  =  ( ( abs `  K )  x.  ( abs `  M
) ) )
16 absmul 11779 . . . . . . 7  |-  ( ( K  e.  CC  /\  N  e.  CC )  ->  ( abs `  ( K  x.  N )
)  =  ( ( abs `  K )  x.  ( abs `  N
) ) )
1715, 16oveqan12d 5877 . . . . . 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 1237 . . . . 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 1224 . . . 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 10066 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
21 zmulcl 10066 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
22 gcdabs 12712 . . . . . 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 463 . . . . 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 1237 . . . 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 11797 . . . . 5  |-  ( K  e.  ZZ  ->  ( abs `  K )  e. 
NN0 )
26 nn0abscl 11797 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  M )  e. 
NN0 )
2726nn0zd 10115 . . . . 5  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
28 nn0abscl 11797 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
2928nn0zd 10115 . . . . 5  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
30 mulgcd 12725 . . . . 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 1224 . . . 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 2323 . . 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 12712 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
34333adant1 973 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
3534oveq2d 5874 . . 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 2315 . 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 2326 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735    x. cmul 8742   NN0cn0 9965   ZZcz 10024   abscabs 11719    gcd cgcd 12685
This theorem is referenced by:  coprmdvds  12781
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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