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Theorem rplpwr 12782
Description: If  A and  B are relatively prime, then so are  A ^ N and  B. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rplpwr  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ N )  gcd  B
)  =  1 ) )

Proof of Theorem rplpwr
Dummy variables  n  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5908 . . . . . . . 8  |-  ( k  =  1  ->  ( A ^ k )  =  ( A ^ 1 ) )
21oveq1d 5915 . . . . . . 7  |-  ( k  =  1  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ 1 )  gcd 
B ) )
32eqeq1d 2324 . . . . . 6  |-  ( k  =  1  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ 1 )  gcd  B )  =  1 ) )
43imbi2d 307 . . . . 5  |-  ( k  =  1  ->  (
( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ k
)  gcd  B )  =  1 )  <->  ( (
( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  ( ( A ^ 1 )  gcd 
B )  =  1 ) ) )
5 oveq2 5908 . . . . . . . 8  |-  ( k  =  n  ->  ( A ^ k )  =  ( A ^ n
) )
65oveq1d 5915 . . . . . . 7  |-  ( k  =  n  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ n )  gcd 
B ) )
76eqeq1d 2324 . . . . . 6  |-  ( k  =  n  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ n
)  gcd  B )  =  1 ) )
87imbi2d 307 . . . . 5  |-  ( k  =  n  ->  (
( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ k
)  gcd  B )  =  1 )  <->  ( (
( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  ( ( A ^ n )  gcd 
B )  =  1 ) ) )
9 oveq2 5908 . . . . . . . 8  |-  ( k  =  ( n  + 
1 )  ->  ( A ^ k )  =  ( A ^ (
n  +  1 ) ) )
109oveq1d 5915 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ ( n  + 
1 ) )  gcd 
B ) )
1110eqeq1d 2324 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ (
n  +  1 ) )  gcd  B )  =  1 ) )
1211imbi2d 307 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ k
)  gcd  B )  =  1 )  <->  ( (
( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  ( ( A ^ ( n  + 
1 ) )  gcd 
B )  =  1 ) ) )
13 oveq2 5908 . . . . . . . 8  |-  ( k  =  N  ->  ( A ^ k )  =  ( A ^ N
) )
1413oveq1d 5915 . . . . . . 7  |-  ( k  =  N  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ N )  gcd 
B ) )
1514eqeq1d 2324 . . . . . 6  |-  ( k  =  N  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ N
)  gcd  B )  =  1 ) )
1615imbi2d 307 . . . . 5  |-  ( k  =  N  ->  (
( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ k
)  gcd  B )  =  1 )  <->  ( (
( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  ( ( A ^ N )  gcd 
B )  =  1 ) ) )
17 nncn 9799 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  CC )
1817exp1d 11287 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( A ^ 1 )  =  A )
1918oveq1d 5915 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( A ^ 1 )  gcd  B )  =  ( A  gcd  B ) )
2019adantr 451 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A ^
1 )  gcd  B
)  =  ( A  gcd  B ) )
2120eqeq1d 2324 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( ( A ^ 1 )  gcd 
B )  =  1  <-> 
( A  gcd  B
)  =  1 ) )
2221biimpar 471 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  ( ( A ^ 1 )  gcd 
B )  =  1 )
23 df-3an 936 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  <->  ( ( A  e.  NN  /\  B  e.  NN )  /\  n  e.  NN ) )
24 simpl1 958 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2524nncnd 9807 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  CC )
26 simpl3 960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN )
2726nnnn0d 10065 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN0 )
2825, 27expp1d 11293 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( ( A ^ n
)  x.  A ) )
29 simp1 955 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  NN )
30 nnnn0 10019 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  n  e.  NN0 )
31303ad2ant3 978 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  n  e.  NN0 )
3229, 31nnexpcld 11313 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  NN )
3332nnzd 10163 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  ZZ )
3433adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  ZZ )
3534zcnd 10165 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  CC )
3635, 25mulcomd 8901 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ n )  x.  A )  =  ( A  x.  ( A ^ n ) ) )
3728, 36eqtrd 2348 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( A  x.  ( A ^ n ) ) )
3837oveq2d 5916 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  ( A ^ ( n  +  1 ) ) )  =  ( B  gcd  ( A  x.  ( A ^ n ) ) ) )
39 simpl2 959 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  NN )
4032adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  NN )
41 nnz 10092 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  NN  ->  A  e.  ZZ )
42413ad2ant1 976 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  ZZ )
43 nnz 10092 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN  ->  B  e.  ZZ )
44433ad2ant2 977 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  B  e.  ZZ )
45 gcdcom 12746 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
4642, 44, 45syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
4746eqeq1d 2324 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  (
( A  gcd  B
)  =  1  <->  ( B  gcd  A )  =  1 ) )
4847biimpa 470 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  A )  =  1 )
49 rpmulgcd 12781 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  NN  /\  A  e.  NN  /\  ( A ^ n )  e.  NN )  /\  ( B  gcd  A )  =  1 )  -> 
( B  gcd  ( A  x.  ( A ^ n ) ) )  =  ( B  gcd  ( A ^
n ) ) )
5039, 24, 40, 48, 49syl31anc 1185 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  ( A  x.  ( A ^ n ) ) )  =  ( B  gcd  ( A ^
n ) ) )
5138, 50eqtrd 2348 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  ( A ^ ( n  +  1 ) ) )  =  ( B  gcd  ( A ^
n ) ) )
52 peano2nn 9803 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
53523ad2ant3 978 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  (
n  +  1 )  e.  NN )
5453adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN )
5554nnnn0d 10065 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN0 )
5624, 55nnexpcld 11313 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  NN )
5756nnzd 10163 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  ZZ )
5844adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  ZZ )
59 gcdcom 12746 . . . . . . . . . . . . 13  |-  ( ( ( A ^ (
n  +  1 ) )  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
( n  +  1 ) )  gcd  B
)  =  ( B  gcd  ( A ^
( n  +  1 ) ) ) )
6057, 58, 59syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ ( n  + 
1 ) )  gcd 
B )  =  ( B  gcd  ( A ^ ( n  + 
1 ) ) ) )
61 gcdcom 12746 . . . . . . . . . . . . 13  |-  ( ( ( A ^ n
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
n )  gcd  B
)  =  ( B  gcd  ( A ^
n ) ) )
6234, 58, 61syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ n )  gcd 
B )  =  ( B  gcd  ( A ^ n ) ) )
6351, 60, 623eqtr4d 2358 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ ( n  + 
1 ) )  gcd 
B )  =  ( ( A ^ n
)  gcd  B )
)
6463eqeq1d 2324 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( ( A ^ ( n  +  1 ) )  gcd  B )  =  1  <->  ( ( A ^ n )  gcd 
B )  =  1 ) )
6564biimprd 214 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( ( A ^ n )  gcd  B )  =  1  ->  ( ( A ^ ( n  + 
1 ) )  gcd 
B )  =  1 ) )
6623, 65sylanbr 459 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  n  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( ( A ^
n )  gcd  B
)  =  1  -> 
( ( A ^
( n  +  1 ) )  gcd  B
)  =  1 ) )
6766an32s 779 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  /\  n  e.  NN )  ->  (
( ( A ^
n )  gcd  B
)  =  1  -> 
( ( A ^
( n  +  1 ) )  gcd  B
)  =  1 ) )
6867expcom 424 . . . . . 6  |-  ( n  e.  NN  ->  (
( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( ( A ^
n )  gcd  B
)  =  1  -> 
( ( A ^
( n  +  1 ) )  gcd  B
)  =  1 ) ) )
6968a2d 23 . . . . 5  |-  ( n  e.  NN  ->  (
( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ n
)  gcd  B )  =  1 )  -> 
( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ (
n  +  1 ) )  gcd  B )  =  1 ) ) )
704, 8, 12, 16, 22, 69nnind 9809 . . . 4  |-  ( N  e.  NN  ->  (
( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ N
)  gcd  B )  =  1 ) )
7170exp3a 425 . . 3  |-  ( N  e.  NN  ->  (
( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ N )  gcd 
B )  =  1 ) ) )
7271com12 27 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( N  e.  NN  ->  ( ( A  gcd  B )  =  1  -> 
( ( A ^ N )  gcd  B
)  =  1 ) ) )
73723impia 1148 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ N )  gcd  B
)  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701  (class class class)co 5900   1c1 8783    + caddc 8785    x. cmul 8787   NNcn 9791   NN0cn0 10012   ZZcz 10071   ^cexp 11151    gcd cgcd 12732
This theorem is referenced by:  rppwr  12783  lgsne0  20625  2sqlem8  20664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fl 10972  df-mod 11021  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-dvds 12579  df-gcd 12733
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