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Theorem rppwr 14045
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 994 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2 simpl2 995 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  NN )
3 simpl3 996 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  N  e.  NN )
43nnnn0d 10843 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  N  e.  NN0 )
52, 4nnexpcld 12288 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B ^ N )  e.  NN )
61, 5, 33jca 1171 . . 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 10956 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  ZZ )
85nnzd 10956 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B ^ N )  e.  ZZ )
9 gcdcom 14008 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( B ^ N )  e.  ZZ )  -> 
( A  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd 
A ) )
107, 8, 9syl2anc 661 . . . 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 1171 . . . . 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 10877 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
13123ad2ant1 1012 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  A  e.  ZZ )
14 nnz 10877 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
15143ad2ant2 1013 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  B  e.  ZZ )
16 gcdcom 14008 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
1713, 15, 16syl2anc 661 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
1817eqeq1d 2464 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  <->  ( B  gcd  A )  =  1 ) )
1918biimpa 484 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  A )  =  1 )
20 rplpwr 14044 . . . . 5  |-  ( ( B  e.  NN  /\  A  e.  NN  /\  N  e.  NN )  ->  (
( B  gcd  A
)  =  1  -> 
( ( B ^ N )  gcd  A
)  =  1 ) )
2111, 19, 20sylc 60 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( B ^ N )  gcd 
A )  =  1 )
2210, 21eqtrd 2503 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A  gcd  ( B ^ N ) )  =  1 )
23 rplpwr 14044 . . 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 60 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ N )  gcd  ( B ^ N
) )  =  1 )
2524ex 434 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ N )  gcd  ( B ^ N ) )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762  (class class class)co 6277   1c1 9484   NNcn 10527   ZZcz 10855   ^cexp 12124    gcd cgcd 13994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-dvds 13839  df-gcd 13995
This theorem is referenced by:  sqgcd  14046  ostth3  23546
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