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Theorem rppwr 12985
Description: If  A and  B are relatively prime, then so are  A ^ N and  B ^ N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rppwr  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ N )  gcd  ( B ^ N ) )  =  1 ) )

Proof of Theorem rppwr
StepHypRef Expression
1 simpl1 960 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2 simpl2 961 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  NN )
3 simpl3 962 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  N  e.  NN )
43nnnn0d 10207 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  N  e.  NN0 )
52, 4nnexpcld 11472 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B ^ N )  e.  NN )
61, 5, 33jca 1134 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A  e.  NN  /\  ( B ^ N )  e.  NN  /\  N  e.  NN ) )
71nnzd 10307 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  ZZ )
85nnzd 10307 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B ^ N )  e.  ZZ )
9 gcdcom 12948 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( B ^ N )  e.  ZZ )  -> 
( A  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd 
A ) )
107, 8, 9syl2anc 643 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd  A ) )
112, 1, 33jca 1134 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  e.  NN  /\  A  e.  NN  /\  N  e.  NN ) )
12 nnz 10236 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
13123ad2ant1 978 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  A  e.  ZZ )
14 nnz 10236 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
15143ad2ant2 979 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  B  e.  ZZ )
16 gcdcom 12948 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
1713, 15, 16syl2anc 643 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
1817eqeq1d 2396 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  <->  ( B  gcd  A )  =  1 ) )
1918biimpa 471 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  A )  =  1 )
20 rplpwr 12984 . . . . 5  |-  ( ( B  e.  NN  /\  A  e.  NN  /\  N  e.  NN )  ->  (
( B  gcd  A
)  =  1  -> 
( ( B ^ N )  gcd  A
)  =  1 ) )
2111, 19, 20sylc 58 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( B ^ N )  gcd 
A )  =  1 )
2210, 21eqtrd 2420 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A  gcd  ( B ^ N ) )  =  1 )
23 rplpwr 12984 . . 3  |-  ( ( A  e.  NN  /\  ( B ^ N )  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  ( B ^ N ) )  =  1  ->  (
( A ^ N
)  gcd  ( B ^ N ) )  =  1 ) )
246, 22, 23sylc 58 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ N )  gcd  ( B ^ N
) )  =  1 )
2524ex 424 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ N )  gcd  ( B ^ N ) )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6021   1c1 8925   NNcn 9933   ZZcz 10215   ^cexp 11310    gcd cgcd 12934
This theorem is referenced by:  sqgcd  12986  ostth3  21200
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-dvds 12781  df-gcd 12935
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