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Theorem rplpwr 10623
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 5571 . . . . . . . 8  |-  ( k  =  1  ->  ( A ^ k )  =  ( A ^ 1 ) )
21oveq1d 5578 . . . . . . 7  |-  ( k  =  1  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ 1 )  gcd 
B ) )
32eqeq1d 2091 . . . . . 6  |-  ( k  =  1  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ 1 )  gcd  B )  =  1 ) )
43imbi2d 228 . . . . 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 5571 . . . . . . . 8  |-  ( k  =  n  ->  ( A ^ k )  =  ( A ^ n
) )
65oveq1d 5578 . . . . . . 7  |-  ( k  =  n  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ n )  gcd 
B ) )
76eqeq1d 2091 . . . . . 6  |-  ( k  =  n  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ n
)  gcd  B )  =  1 ) )
87imbi2d 228 . . . . 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 5571 . . . . . . . 8  |-  ( k  =  ( n  + 
1 )  ->  ( A ^ k )  =  ( A ^ (
n  +  1 ) ) )
109oveq1d 5578 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ ( n  + 
1 ) )  gcd 
B ) )
1110eqeq1d 2091 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ (
n  +  1 ) )  gcd  B )  =  1 ) )
1211imbi2d 228 . . . . 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 5571 . . . . . . . 8  |-  ( k  =  N  ->  ( A ^ k )  =  ( A ^ N
) )
1413oveq1d 5578 . . . . . . 7  |-  ( k  =  N  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ N )  gcd 
B ) )
1514eqeq1d 2091 . . . . . 6  |-  ( k  =  N  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ N
)  gcd  B )  =  1 ) )
1615imbi2d 228 . . . . 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 8166 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  CC )
1817exp1d 9749 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( A ^ 1 )  =  A )
1918oveq1d 5578 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( A ^ 1 )  gcd  B )  =  ( A  gcd  B ) )
2019adantr 270 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A ^
1 )  gcd  B
)  =  ( A  gcd  B ) )
2120eqeq1d 2091 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( ( A ^ 1 )  gcd 
B )  =  1  <-> 
( A  gcd  B
)  =  1 ) )
2221biimpar 291 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  ( ( A ^ 1 )  gcd 
B )  =  1 )
23 df-3an 922 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  <->  ( ( A  e.  NN  /\  B  e.  NN )  /\  n  e.  NN ) )
24 simpl1 942 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2524nncnd 8172 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  CC )
26 simpl3 944 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN )
2726nnnn0d 8460 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN0 )
2825, 27expp1d 9755 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( ( A ^ n
)  x.  A ) )
29 simp1 939 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  NN )
30 nnnn0 8414 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  n  e.  NN0 )
31303ad2ant3 962 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  n  e.  NN0 )
3229, 31nnexpcld 9776 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  NN )
3332nnzd 8601 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  ZZ )
3433adantr 270 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  ZZ )
3534zcnd 8603 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  CC )
3635, 25mulcomd 7254 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ n )  x.  A )  =  ( A  x.  ( A ^ n ) ) )
3728, 36eqtrd 2115 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( A  x.  ( A ^ n ) ) )
3837oveq2d 5579 . . . . . . . . . . . . 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 943 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  NN )
4032adantr 270 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  NN )
41 nnz 8503 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  NN  ->  A  e.  ZZ )
42413ad2ant1 960 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  ZZ )
43 nnz 8503 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN  ->  B  e.  ZZ )
44433ad2ant2 961 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  B  e.  ZZ )
45 gcdcom 10572 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
4642, 44, 45syl2anc 403 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
4746eqeq1d 2091 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  (
( A  gcd  B
)  =  1  <->  ( B  gcd  A )  =  1 ) )
4847biimpa 290 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  A )  =  1 )
49 rpmulgcd 10622 . . . . . . . . . . . . . 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 1173 . . . . . . . . . . . . 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 2115 . . . . . . . . . . . 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 8170 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
53523ad2ant3 962 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  (
n  +  1 )  e.  NN )
5453adantr 270 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN )
5554nnnn0d 8460 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN0 )
5624, 55nnexpcld 9776 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  NN )
5756nnzd 8601 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  ZZ )
5844adantr 270 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  ZZ )
59 gcdcom 10572 . . . . . . . . . . . . 13  |-  ( ( ( A ^ (
n  +  1 ) )  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
( n  +  1 ) )  gcd  B
)  =  ( B  gcd  ( A ^
( n  +  1 ) ) ) )
6057, 58, 59syl2anc 403 . . . . . . . . . . . 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 10572 . . . . . . . . . . . . 13  |-  ( ( ( A ^ n
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
n )  gcd  B
)  =  ( B  gcd  ( A ^
n ) ) )
6234, 58, 61syl2anc 403 . . . . . . . . . . . 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 2125 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ ( n  + 
1 ) )  gcd 
B )  =  ( ( A ^ n
)  gcd  B )
)
6463eqeq1d 2091 . . . . . . . . . 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 156 . . . . . . . . 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 279 . . . . . . . 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 533 . . . . . . 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 114 . . . . . 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 26 . . . . 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 8174 . . . 4  |-  ( N  e.  NN  ->  (
( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ N
)  gcd  B )  =  1 ) )
7170expd 254 . . 3  |-  ( N  e.  NN  ->  (
( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ N )  gcd 
B )  =  1 ) ) )
7271com12 30 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( N  e.  NN  ->  ( ( A  gcd  B )  =  1  -> 
( ( A ^ N )  gcd  B
)  =  1 ) ) )
73723impia 1136 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5563   1c1 7096    + caddc 7098    x. cmul 7100   NNcn 8158   NN0cn0 8407   ZZcz 8484   ^cexp 9624    gcd cgcd 10545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546
This theorem is referenced by:  rppwr  10624
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