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Theorem rplpwr 13736
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 6098 . . . . . . . 8  |-  ( k  =  1  ->  ( A ^ k )  =  ( A ^ 1 ) )
21oveq1d 6105 . . . . . . 7  |-  ( k  =  1  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ 1 )  gcd 
B ) )
32eqeq1d 2449 . . . . . 6  |-  ( k  =  1  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ 1 )  gcd  B )  =  1 ) )
43imbi2d 316 . . . . 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 6098 . . . . . . . 8  |-  ( k  =  n  ->  ( A ^ k )  =  ( A ^ n
) )
65oveq1d 6105 . . . . . . 7  |-  ( k  =  n  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ n )  gcd 
B ) )
76eqeq1d 2449 . . . . . 6  |-  ( k  =  n  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ n
)  gcd  B )  =  1 ) )
87imbi2d 316 . . . . 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 6098 . . . . . . . 8  |-  ( k  =  ( n  + 
1 )  ->  ( A ^ k )  =  ( A ^ (
n  +  1 ) ) )
109oveq1d 6105 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ ( n  + 
1 ) )  gcd 
B ) )
1110eqeq1d 2449 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ (
n  +  1 ) )  gcd  B )  =  1 ) )
1211imbi2d 316 . . . . 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 6098 . . . . . . . 8  |-  ( k  =  N  ->  ( A ^ k )  =  ( A ^ N
) )
1413oveq1d 6105 . . . . . . 7  |-  ( k  =  N  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ N )  gcd 
B ) )
1514eqeq1d 2449 . . . . . 6  |-  ( k  =  N  ->  (
( ( A ^
k )  gcd  B
)  =  1  <->  (
( A ^ N
)  gcd  B )  =  1 ) )
1615imbi2d 316 . . . . 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 10326 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  CC )
1817exp1d 11999 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( A ^ 1 )  =  A )
1918oveq1d 6105 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( A ^ 1 )  gcd  B )  =  ( A  gcd  B ) )
2019adantr 462 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A ^
1 )  gcd  B
)  =  ( A  gcd  B ) )
2120eqeq1d 2449 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( ( A ^ 1 )  gcd 
B )  =  1  <-> 
( A  gcd  B
)  =  1 ) )
2221biimpar 482 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  ( ( A ^ 1 )  gcd 
B )  =  1 )
23 df-3an 962 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  <->  ( ( A  e.  NN  /\  B  e.  NN )  /\  n  e.  NN ) )
24 simpl1 986 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2524nncnd 10334 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  CC )
26 simpl3 988 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN )
2726nnnn0d 10632 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN0 )
2825, 27expp1d 12005 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( ( A ^ n
)  x.  A ) )
29 simp1 983 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  NN )
30 nnnn0 10582 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  n  e.  NN0 )
31303ad2ant3 1006 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  n  e.  NN0 )
3229, 31nnexpcld 12025 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  NN )
3332nnzd 10742 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  ZZ )
3433adantr 462 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  ZZ )
3534zcnd 10744 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  CC )
3635, 25mulcomd 9403 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ n )  x.  A )  =  ( A  x.  ( A ^ n ) ) )
3728, 36eqtrd 2473 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( A  x.  ( A ^ n ) ) )
3837oveq2d 6106 . . . . . . . . . . . . 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 987 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  NN )
4032adantr 462 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  NN )
41 nnz 10664 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  NN  ->  A  e.  ZZ )
42413ad2ant1 1004 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  ZZ )
43 nnz 10664 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN  ->  B  e.  ZZ )
44433ad2ant2 1005 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  B  e.  ZZ )
45 gcdcom 13700 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
4642, 44, 45syl2anc 656 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
4746eqeq1d 2449 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  (
( A  gcd  B
)  =  1  <->  ( B  gcd  A )  =  1 ) )
4847biimpa 481 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  A )  =  1 )
49 rpmulgcd 13735 . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . 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 2473 . . . . . . . . . . . 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 10330 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
53523ad2ant3 1006 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  (
n  +  1 )  e.  NN )
5453adantr 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN )
5554nnnn0d 10632 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN0 )
5624, 55nnexpcld 12025 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  NN )
5756nnzd 10742 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  ZZ )
5844adantr 462 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  ZZ )
59 gcdcom 13700 . . . . . . . . . . . . 13  |-  ( ( ( A ^ (
n  +  1 ) )  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
( n  +  1 ) )  gcd  B
)  =  ( B  gcd  ( A ^
( n  +  1 ) ) ) )
6057, 58, 59syl2anc 656 . . . . . . . . . . . 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 13700 . . . . . . . . . . . . 13  |-  ( ( ( A ^ n
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
n )  gcd  B
)  =  ( B  gcd  ( A ^
n ) ) )
6234, 58, 61syl2anc 656 . . . . . . . . . . . 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 2483 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ ( n  + 
1 ) )  gcd 
B )  =  ( ( A ^ n
)  gcd  B )
)
6463eqeq1d 2449 . . . . . . . . . 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 223 . . . . . . . . 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 470 . . . . . . . 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 797 . . . . . . 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 435 . . . . . 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 10336 . . . 4  |-  ( N  e.  NN  ->  (
( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ N
)  gcd  B )  =  1 ) )
7170exp3a 436 . . 3  |-  ( N  e.  NN  ->  (
( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ N )  gcd 
B )  =  1 ) ) )
7271com12 31 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( N  e.  NN  ->  ( ( A  gcd  B )  =  1  -> 
( ( A ^ N )  gcd  B
)  =  1 ) ) )
73723impia 1179 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  N  e.  NN )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ N )  gcd  B
)  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761  (class class class)co 6090   1c1 9279    + caddc 9281    x. cmul 9283   NNcn 10318   NN0cn0 10575   ZZcz 10642   ^cexp 11861    gcd cgcd 13686
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687
This theorem is referenced by:  rppwr  13737  lgsne0  22631  2sqlem8  22670
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