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Theorem absmulgcd 11433
Description: Distribute absolute value of multiplication over gcd. Theorem 1.4(c) in [ApostolNT] p. 16. (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 11385 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
2 nn0re 8780 . . . . . 6  |-  ( ( M  gcd  N )  e.  NN0  ->  ( M  gcd  N )  e.  RR )
3 nn0ge0 8796 . . . . . 6  |-  ( ( M  gcd  N )  e.  NN0  ->  0  <_ 
( M  gcd  N
) )
42, 3absidd 10715 . . . . 5  |-  ( ( M  gcd  N )  e.  NN0  ->  ( abs `  ( M  gcd  N
) )  =  ( M  gcd  N ) )
51, 4syl 14 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( abs `  ( M  gcd  N ) )  =  ( M  gcd  N ) )
65oveq2d 5706 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  K
)  x.  ( abs `  ( M  gcd  N
) ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
763adant1 964 . 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 8853 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  CC )
91nn0cnd 8826 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  CC )
10 absmul 10617 . . . 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 284 . . 3  |-  ( ( K  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  ( K  x.  ( M  gcd  N ) ) )  =  ( ( abs `  K )  x.  ( abs `  ( M  gcd  N ) ) ) )
12113impb 1142 . 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 8853 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 zcn 8853 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  CC )
15 absmul 10617 . . . . . . 7  |-  ( ( K  e.  CC  /\  M  e.  CC )  ->  ( abs `  ( K  x.  M )
)  =  ( ( abs `  K )  x.  ( abs `  M
) ) )
16 absmul 10617 . . . . . . 7  |-  ( ( K  e.  CC  /\  N  e.  CC )  ->  ( abs `  ( K  x.  N )
)  =  ( ( abs `  K )  x.  ( abs `  N
) ) )
1715, 16oveqan12d 5709 . . . . . 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 1236 . . . . 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 1223 . . . 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 8901 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
21 zmulcl 8901 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
22 gcdabs 11406 . . . . . 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 284 . . . . 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 1236 . . . 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 10633 . . . . 5  |-  ( K  e.  ZZ  ->  ( abs `  K )  e. 
NN0 )
26 zabscl 10634 . . . . 5  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
27 zabscl 10634 . . . . 5  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
28 mulgcd 11432 . . . . 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 ) ) ) )
2925, 26, 27, 28syl3an 1223 . . . 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 ) ) ) )
3019, 24, 293eqtr3d 2135 . . 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
) ) ) )
31 gcdabs 11406 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
32313adant1 964 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  M
)  gcd  ( abs `  N ) )  =  ( M  gcd  N
) )
3332oveq2d 5706 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( abs `  K
)  x.  ( ( abs `  M )  gcd  ( abs `  N
) ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
3430, 33eqtrd 2127 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  gcd  ( K  x.  N ) )  =  ( ( abs `  K
)  x.  ( M  gcd  N ) ) )
357, 12, 343eqtr4rd 2138 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 103    /\ w3a 927    = wceq 1296    e. wcel 1445   ` cfv 5049  (class class class)co 5690   CCcc 7445    x. cmul 7452   NN0cn0 8771   ZZcz 8848   abscabs 10545    gcd cgcd 11365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-q 9204  df-rp 9234  df-fz 9574  df-fzo 9703  df-fl 9826  df-mod 9879  df-seqfrec 10001  df-exp 10070  df-cj 10391  df-re 10392  df-im 10393  df-rsqrt 10546  df-abs 10547  df-dvds 11224  df-gcd 11366
This theorem is referenced by: (None)
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