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Theorem rppwr 14294
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 998 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2 simpl2 999 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  NN )
3 simpl3 1000 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  N  e.  NN )
43nnnn0d 10811 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  N  e.  NN0 )
52, 4nnexpcld 12283 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B ^ N )  e.  NN )
61, 5, 33jca 1175 . . 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 10925 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  ZZ )
85nnzd 10925 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B ^ N )  e.  ZZ )
9 gcdcom 14257 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( B ^ N )  e.  ZZ )  -> 
( A  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd 
A ) )
107, 8, 9syl2anc 659 . . . 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 1175 . . . . 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 10845 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
13123ad2ant1 1016 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  A  e.  ZZ )
14 nnz 10845 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
15143ad2ant2 1017 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  B  e.  ZZ )
16 gcdcom 14257 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
1713, 15, 16syl2anc 659 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
1817eqeq1d 2402 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  <->  ( B  gcd  A )  =  1 ) )
1918biimpa 482 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  A )  =  1 )
20 rplpwr 14293 . . . . 5  |-  ( ( B  e.  NN  /\  A  e.  NN  /\  N  e.  NN )  ->  (
( B  gcd  A
)  =  1  -> 
( ( B ^ N )  gcd  A
)  =  1 ) )
2111, 19, 20sylc 59 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( B ^ N )  gcd 
A )  =  1 )
2210, 21eqtrd 2441 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A  gcd  ( B ^ N ) )  =  1 )
23 rplpwr 14293 . . 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 59 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ N )  gcd  ( B ^ N
) )  =  1 )
2524ex 432 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840  (class class class)co 6232   1c1 9441   NNcn 10494   ZZcz 10823   ^cexp 12118    gcd cgcd 14243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-2nd 6737  df-recs 6997  df-rdg 7031  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-fl 11877  df-mod 11946  df-seq 12060  df-exp 12119  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-dvds 14086  df-gcd 14244
This theorem is referenced by:  sqgcd  14295  ostth3  24094
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