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Theorem rpmulgcd 11293
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 11287 . . . . . 6  |-  ( ( K  e.  NN  /\  N  e.  NN )  ->  ( K  gcd  ( K  x.  N )
)  =  K )
213adant2 962 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( K  gcd  ( K  x.  N ) )  =  K )
32oveq1d 5667 . . . 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 8769 . . . . . 6  |-  ( K  e.  NN  ->  K  e.  ZZ )
543ad2ant1 964 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  K  e.  ZZ )
6 nnz 8769 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  ZZ )
7 zmulcl 8803 . . . . . . 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 962 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( K  x.  N )  e.  ZZ )
10 nnz 8769 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  ZZ )
11 zmulcl 8803 . . . . . . 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 961 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( M  x.  N )  e.  ZZ )
14 gcdass 11282 . . . . 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 1174 . . . 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 2122 . . 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 8680 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
19 mulgcdr 11285 . . . . . 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 1216 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( K  x.  N
)  gcd  ( M  x.  N ) )  =  ( ( K  gcd  M )  x.  N ) )
21 oveq1 5659 . . . . 5  |-  ( ( K  gcd  M )  =  1  ->  (
( K  gcd  M
)  x.  N )  =  ( 1  x.  N ) )
2220, 21sylan9eq 2140 . . . 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 8430 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
24233ad2ant3 966 . . . . . 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 7506 . . . 4  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( 1  x.  N )  =  N )
2722, 26eqtrd 2120 . . 3  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( K  x.  N )  gcd  ( M  x.  N
) )  =  N )
2827oveq2d 5668 . 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 2120 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 924    = wceq 1289    e. wcel 1438  (class class class)co 5652   CCcc 7348   1c1 7351    x. cmul 7355   NNcn 8422   NN0cn0 8673   ZZcz 8750    gcd cgcd 11216
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-sup 6679  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-fl 9677  df-mod 9730  df-iseq 9853  df-seq3 9854  df-exp 9955  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-dvds 11075  df-gcd 11217
This theorem is referenced by:  rplpwr  11294
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