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Theorem rppwr 12736
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 958 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2 simpl2 959 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  NN )
3 simpl3 960 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  N  e.  NN )
43nnnn0d 10018 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  N  e.  NN0 )
52, 4nnexpcld 11266 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B ^ N )  e.  NN )
61, 5, 33jca 1132 . . 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 10116 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  ZZ )
85nnzd 10116 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B ^ N )  e.  ZZ )
9 gcdcom 12699 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( B ^ N )  e.  ZZ )  -> 
( A  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd 
A ) )
107, 8, 9syl2anc 642 . . . 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 1132 . . . . 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 10045 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
13123ad2ant1 976 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  A  e.  ZZ )
14 nnz 10045 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
15143ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  B  e.  ZZ )
16 gcdcom 12699 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
1713, 15, 16syl2anc 642 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
1817eqeq1d 2291 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  <->  ( B  gcd  A )  =  1 ) )
1918biimpa 470 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  A )  =  1 )
20 rplpwr 12735 . . . . 5  |-  ( ( B  e.  NN  /\  A  e.  NN  /\  N  e.  NN )  ->  (
( B  gcd  A
)  =  1  -> 
( ( B ^ N )  gcd  A
)  =  1 ) )
2111, 19, 20sylc 56 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( B ^ N )  gcd 
A )  =  1 )
2210, 21eqtrd 2315 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A  gcd  ( B ^ N ) )  =  1 )
23 rplpwr 12735 . . 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 56 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ N )  gcd  ( B ^ N
) )  =  1 )
2524ex 423 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858   1c1 8738   NNcn 9746   ZZcz 10024   ^cexp 11104    gcd cgcd 12685
This theorem is referenced by:  sqgcd  12737  ostth3  20787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
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