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Theorem rpmulgcd 12982
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 12977 . . . . . 6  |-  ( ( K  e.  NN  /\  N  e.  NN )  ->  ( K  gcd  ( K  x.  N )
)  =  K )
213adant2 976 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( K  gcd  ( K  x.  N ) )  =  K )
32oveq1d 6035 . . . 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 10235 . . . . . 6  |-  ( K  e.  NN  ->  K  e.  ZZ )
543ad2ant1 978 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  K  e.  ZZ )
6 nnz 10235 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  ZZ )
7 zmulcl 10256 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  x.  N
)  e.  ZZ )
84, 6, 7syl2an 464 . . . . . 6  |-  ( ( K  e.  NN  /\  N  e.  NN )  ->  ( K  x.  N
)  e.  ZZ )
983adant2 976 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( K  x.  N )  e.  ZZ )
10 nnz 10235 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  ZZ )
11 zmulcl 10256 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
1210, 6, 11syl2an 464 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  x.  N
)  e.  ZZ )
13123adant1 975 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  ( M  x.  N )  e.  ZZ )
14 gcdass 12972 . . . . 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 1184 . . . 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 2421 . . 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 452 . 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 10160 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
19 mulgcdr 12975 . . . . . 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 1226 . . . . 5  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  (
( K  x.  N
)  gcd  ( M  x.  N ) )  =  ( ( K  gcd  M )  x.  N ) )
21 oveq1 6027 . . . . 5  |-  ( ( K  gcd  M )  =  1  ->  (
( K  gcd  M
)  x.  N )  =  ( 1  x.  N ) )
2220, 21sylan9eq 2439 . . . 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 9940 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
24233ad2ant3 980 . . . . . 6  |-  ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  ->  N  e.  CC )
2524adantr 452 . . . . 5  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  N  e.  CC )
2625mulid2d 9039 . . . 4  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( 1  x.  N )  =  N )
2722, 26eqtrd 2419 . . 3  |-  ( ( ( K  e.  NN  /\  M  e.  NN  /\  N  e.  NN )  /\  ( K  gcd  M
)  =  1 )  ->  ( ( K  x.  N )  gcd  ( M  x.  N
) )  =  N )
2827oveq2d 6036 . 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 2419 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6020   CCcc 8921   1c1 8924    x. cmul 8928   NNcn 9932   NN0cn0 10153   ZZcz 10214    gcd cgcd 12933
This theorem is referenced by:  rplpwr  12983  lgsquad2lem2  21010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-dvds 12780  df-gcd 12934
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