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Theorem rplpwr 14042
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 6283 . . . . . . . 8  |-  ( k  =  1  ->  ( A ^ k )  =  ( A ^ 1 ) )
21oveq1d 6290 . . . . . . 7  |-  ( k  =  1  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ 1 )  gcd 
B ) )
32eqeq1d 2462 . . . . . 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 6283 . . . . . . . 8  |-  ( k  =  n  ->  ( A ^ k )  =  ( A ^ n
) )
65oveq1d 6290 . . . . . . 7  |-  ( k  =  n  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ n )  gcd 
B ) )
76eqeq1d 2462 . . . . . 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 6283 . . . . . . . 8  |-  ( k  =  ( n  + 
1 )  ->  ( A ^ k )  =  ( A ^ (
n  +  1 ) ) )
109oveq1d 6290 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ ( n  + 
1 ) )  gcd 
B ) )
1110eqeq1d 2462 . . . . . 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 6283 . . . . . . . 8  |-  ( k  =  N  ->  ( A ^ k )  =  ( A ^ N
) )
1413oveq1d 6290 . . . . . . 7  |-  ( k  =  N  ->  (
( A ^ k
)  gcd  B )  =  ( ( A ^ N )  gcd 
B ) )
1514eqeq1d 2462 . . . . . 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 10533 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  CC )
1817exp1d 12260 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( A ^ 1 )  =  A )
1918oveq1d 6290 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( A ^ 1 )  gcd  B )  =  ( A  gcd  B ) )
2019adantr 465 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A ^
1 )  gcd  B
)  =  ( A  gcd  B ) )
2120eqeq1d 2462 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( ( A ^ 1 )  gcd 
B )  =  1  <-> 
( A  gcd  B
)  =  1 ) )
2221biimpar 485 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  ( ( A ^ 1 )  gcd 
B )  =  1 )
23 df-3an 970 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  <->  ( ( A  e.  NN  /\  B  e.  NN )  /\  n  e.  NN ) )
24 simpl1 994 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  NN )
2524nncnd 10541 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  A  e.  CC )
26 simpl3 996 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN )
2726nnnn0d 10841 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  n  e.  NN0 )
2825, 27expp1d 12266 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( ( A ^ n
)  x.  A ) )
29 simp1 991 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  NN )
30 nnnn0 10791 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  n  e.  NN0 )
31303ad2ant3 1014 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  n  e.  NN0 )
3229, 31nnexpcld 12286 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  NN )
3332nnzd 10954 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A ^ n )  e.  ZZ )
3433adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  ZZ )
3534zcnd 10956 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  CC )
3635, 25mulcomd 9606 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ n )  x.  A )  =  ( A  x.  ( A ^ n ) ) )
3728, 36eqtrd 2501 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  =  ( A  x.  ( A ^ n ) ) )
3837oveq2d 6291 . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  NN )
4032adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
n )  e.  NN )
41 nnz 10875 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  NN  ->  A  e.  ZZ )
42413ad2ant1 1012 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  A  e.  ZZ )
43 nnz 10875 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  NN  ->  B  e.  ZZ )
44433ad2ant2 1013 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  B  e.  ZZ )
45 gcdcom 14006 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
4642, 44, 45syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
4746eqeq1d 2462 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  (
( A  gcd  B
)  =  1  <->  ( B  gcd  A )  =  1 ) )
4847biimpa 484 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( B  gcd  A )  =  1 )
49 rpmulgcd 14041 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 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 2501 . . . . . . . . . . . 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 10537 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
n  +  1 )  e.  NN )
53523ad2ant3 1014 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  ->  (
n  +  1 )  e.  NN )
5453adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN )
5554nnnn0d 10841 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( n  + 
1 )  e.  NN0 )
5624, 55nnexpcld 12286 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  NN )
5756nnzd 10954 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( A ^
( n  +  1 ) )  e.  ZZ )
5844adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  B  e.  ZZ )
59 gcdcom 14006 . . . . . . . . . . . . 13  |-  ( ( ( A ^ (
n  +  1 ) )  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
( n  +  1 ) )  gcd  B
)  =  ( B  gcd  ( A ^
( n  +  1 ) ) ) )
6057, 58, 59syl2anc 661 . . . . . . . . . . . 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 14006 . . . . . . . . . . . . 13  |-  ( ( ( A ^ n
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
n )  gcd  B
)  =  ( B  gcd  ( A ^
n ) ) )
6234, 58, 61syl2anc 661 . . . . . . . . . . . 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 2511 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  n  e.  NN )  /\  ( A  gcd  B
)  =  1 )  ->  ( ( A ^ ( n  + 
1 ) )  gcd 
B )  =  ( ( A ^ n
)  gcd  B )
)
6463eqeq1d 2462 . . . . . . . . . 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 473 . . . . . . . 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 802 . . . . . . 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 10543 . . . 4  |-  ( N  e.  NN  ->  (
( ( A  e.  NN  /\  B  e.  NN )  /\  ( A  gcd  B )  =  1 )  ->  (
( A ^ N
)  gcd  B )  =  1 ) )
7170expd 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 1188 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 968    = wceq 1374    e. wcel 1762  (class class class)co 6275   1c1 9482    + caddc 9484    x. cmul 9486   NNcn 10525   NN0cn0 10784   ZZcz 10853   ^cexp 12122    gcd cgcd 13992
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993
This theorem is referenced by:  rppwr  14043  lgsne0  23329  2sqlem8  23368
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