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Theorem rpmulgcd 10606
Description: If  K and  M are relatively prime, then the GCD of  K and  M  x.  N is the GCD of  K and  N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rpmulgcd  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  =  ( K  gcd  N ) )

Proof of Theorem rpmulgcd
StepHypRef Expression
1 gcdmultiple 10600 . . . . . 6  |-  ( ( K  e.  NN  /\  N  e.  NN )  ->  ( K  gcd  ( K  x.  N )
)  =  K )
213adant2 958 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( K  gcd  ( K  x.  N ) )  =  K )
32oveq1d 5579 . . . 4  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( K  gcd  ( K  x.  N )
)  gcd  ( M  x.  N ) )  =  ( K  gcd  ( M  x.  N )
) )
4 nnz 8487 . . . . . 6  |-  ( K  e.  NN  ->  K  e.  ZZ )
543ad2ant1 960 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  K  e.  ZZ )
6 nnz 8487 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  ZZ )
7 zmulcl 8521 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
84, 6, 7syl2an 283 . . . . . 6  |-  ( ( K  e.  NN  /\  N  e.  NN )  ->  ( K  x.  N
)  e.  ZZ )
983adant2 958 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( K  x.  N )  e.  ZZ )
10 nnz 8487 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  ZZ )
11 zmulcl 8521 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
1210, 6, 11syl2an 283 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  x.  N
)  e.  ZZ )
13123adant1 957 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( M  x.  N )  e.  ZZ )
14 gcdass 10595 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( K  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( ( K  gcd  ( K  x.  N ) )  gcd  ( M  x.  N
) )  =  ( K  gcd  ( ( K  x.  N )  gcd  ( M  x.  N ) ) ) )
155, 9, 13, 14syl3anc 1170 . . . 4  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( K  gcd  ( K  x.  N )
)  gcd  ( M  x.  N ) )  =  ( K  gcd  (
( K  x.  N
)  gcd  ( M  x.  N ) ) ) )
163, 15eqtr3d 2117 . . 3  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( K  gcd  ( M  x.  N ) )  =  ( K  gcd  (
( K  x.  N
)  gcd  ( M  x.  N ) ) ) )
1716adantr 270 . 2  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  =  ( K  gcd  ( ( K  x.  N )  gcd  ( M  x.  N ) ) ) )
18 nnnn0 8398 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
19 mulgcdr 10598 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN0 )  ->  (
( K  x.  N
)  gcd  ( M  x.  N ) )  =  ( ( K  gcd  M )  x.  N ) )
204, 10, 18, 19syl3an 1212 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( K  x.  N
)  gcd  ( M  x.  N ) )  =  ( ( K  gcd  M )  x.  N ) )
21 oveq1 5571 . . . . 5  |-  ( ( K  gcd  M )  =  1  ->  (
( K  gcd  M
)  x.  N )  =  ( 1  x.  N ) )
2220, 21sylan9eq 2135 . . . 4  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( K  x.  N )  gcd  ( M  x.  N
) )  =  ( 1  x.  N ) )
23 nncn 8150 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
24233ad2ant3 962 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  N  e.  CC )
2524adantr 270 . . . . 5  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  N  e.  CC )
2625mulid2d 7235 . . . 4  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( 1  x.  N )  =  N )
2722, 26eqtrd 2115 . . 3  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( K  x.  N )  gcd  ( M  x.  N
) )  =  N )
2827oveq2d 5580 . 2  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  gcd  ( ( K  x.  N )  gcd  ( M  x.  N )
) )  =  ( K  gcd  N ) )
2917, 28eqtrd 2115 1  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( K  gcd  ( M  x.  N
) )  =  ( K  gcd  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5564   CCcc 7077   1c1 7080    x. cmul 7084   NNcn 8142   NN0cn0 8391   ZZcz 8468    gcd cgcd 10529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulrcl 7173  ax-addcom 7174  ax-mulcom 7175  ax-addass 7176  ax-mulass 7177  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-1rid 7181  ax-0id 7182  ax-rnegex 7183  ax-precex 7184  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-apti 7189  ax-pre-ltadd 7190  ax-pre-mulgt0 7191  ax-pre-mulext 7192  ax-arch 7193  ax-caucvg 7194
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-if 3370  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-sup 6492  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-reap 7778  df-ap 7785  df-div 7864  df-inn 8143  df-2 8201  df-3 8202  df-4 8203  df-n0 8392  df-z 8469  df-uz 8737  df-q 8822  df-rp 8852  df-fz 9142  df-fzo 9266  df-fl 9388  df-mod 9441  df-iseq 9558  df-iexp 9609  df-cj 9914  df-re 9915  df-im 9916  df-rsqrt 10069  df-abs 10070  df-dvds 10388  df-gcd 10530
This theorem is referenced by:  rplpwr  10607
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