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Theorem rplpwr 11098
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 5642 . . . . . . . 8  |-  ( k  =  1  ->  ( A ^ k )  =  ( A ^ 1 ) )
21oveq1d 5649 . . . . . . 7  |-  ( k  =  1  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ 1 )  gcd 
B ) )
32eqeq1d 2096 . . . . . 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 5642 . . . . . . . 8  |-  ( k  =  n  ->  ( A ^ k )  =  ( A ^ n
) )
65oveq1d 5649 . . . . . . 7  |-  ( k  =  n  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ n )  gcd 
B ) )
76eqeq1d 2096 . . . . . 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 5642 . . . . . . . 8  |-  ( k  =  ( n  + 
1 )  ->  ( A ^ k )  =  ( A ^ (
n  +  1 ) ) )
109oveq1d 5649 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ ( n  + 
1 ) )  gcd 
B ) )
1110eqeq1d 2096 . . . . . 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 5642 . . . . . . . 8  |-  ( k  =  N  ->  ( A ^ k )  =  ( A ^ N
) )
1413oveq1d 5649 . . . . . . 7  |-  ( k  =  N  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ N )  gcd 
B ) )
1514eqeq1d 2096 . . . . . 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 8402 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  CC )
1817exp1d 10046 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( A ^ 1 )  =  A )
1918oveq1d 5649 . . . . . . . 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 2096 . . . . . 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 926 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  <->  ( ( A  e.  NN  /\  B  e.  NN )  /\  n  e.  NN ) )
24 simpl1 946 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2524nncnd 8408 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  CC )
26 simpl3 948 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN )
2726nnnn0d 8696 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN0 )
2825, 27expp1d 10052 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( ( A ^ n
)  x.  A ) )
29 simp1 943 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  NN )
30 nnnn0 8650 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  n  e.  NN0 )
31303ad2ant3 966 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  n  e.  NN0 )
3229, 31nnexpcld 10073 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  NN )
3332nnzd 8837 . . . . . . . . . . . . . . . . . 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 8839 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  CC )
3635, 25mulcomd 7488 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ n )  x.  A )  =  ( A  x.  ( A ^ n ) ) )
3728, 36eqtrd 2120 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( A  x.  ( A ^ n ) ) )
3837oveq2d 5650 . . . . . . . . . . . . 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 947 . . . . . . . . . . . . . 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 8739 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  NN  ->  A  e.  ZZ )
42413ad2ant1 964 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  ZZ )
43 nnz 8739 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN  ->  B  e.  ZZ )
44433ad2ant2 965 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  B  e.  ZZ )
45 gcdcom 11047 . . . . . . . . . . . . . . . . 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 2096 . . . . . . . . . . . . . . 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 11097 . . . . . . . . . . . . . 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 1177 . . . . . . . . . . . . 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 2120 . . . . . . . . . . . 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 8406 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
53523ad2ant3 966 . . . . . . . . . . . . . . . . 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 8696 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN0 )
5624, 55nnexpcld 10073 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  NN )
5756nnzd 8837 . . . . . . . . . . . . 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 11047 . . . . . . . . . . . . 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 11047 . . . . . . . . . . . . 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 2130 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ ( n  + 
1 ) )  gcd 
B )  =  ( ( A ^ n
)  gcd  B )
)
6463eqeq1d 2096 . . . . . . . . . 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 535 . . . . . . 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 8410 . . . 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 1140 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 924    = wceq 1289    e. wcel 1438  (class class class)co 5634   1c1 7330    + caddc 7332    x. cmul 7334   NNcn 8394   NN0cn0 8643   ZZcz 8720   ^cexp 9919    gcd cgcd 11020
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10879  df-gcd 11021
This theorem is referenced by:  rppwr  11099
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