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Theorem pcexp 13238
Description: Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)
Assertion
Ref Expression
pcexp  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  ( P  pCnt  ( A ^ N ) )  =  ( N  x.  ( P  pCnt  A ) ) )

Proof of Theorem pcexp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6092 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
21oveq2d 6100 . . . 4  |-  ( x  =  0  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ 0 ) ) )
3 oveq1 6091 . . . 4  |-  ( x  =  0  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( 0  x.  ( P  pCnt  A ) ) )
42, 3eqeq12d 2452 . . 3  |-  ( x  =  0  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P 
pCnt  A ) ) ) )
5 oveq2 6092 . . . . 5  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
65oveq2d 6100 . . . 4  |-  ( x  =  y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ y ) ) )
7 oveq1 6091 . . . 4  |-  ( x  =  y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( y  x.  ( P  pCnt  A ) ) )
86, 7eqeq12d 2452 . . 3  |-  ( x  =  y  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ y
) )  =  ( y  x.  ( P 
pCnt  A ) ) ) )
9 oveq2 6092 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A ^ x )  =  ( A ^ (
y  +  1 ) ) )
109oveq2d 6100 . . . 4  |-  ( x  =  ( y  +  1 )  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ ( y  +  1 ) ) ) )
11 oveq1 6091 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( ( y  +  1 )  x.  ( P  pCnt  A ) ) )
1210, 11eqeq12d 2452 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ (
y  +  1 ) ) )  =  ( ( y  +  1 )  x.  ( P 
pCnt  A ) ) ) )
13 oveq2 6092 . . . . 5  |-  ( x  =  -u y  ->  ( A ^ x )  =  ( A ^ -u y
) )
1413oveq2d 6100 . . . 4  |-  ( x  =  -u y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ -u y ) ) )
15 oveq1 6091 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( -u y  x.  ( P  pCnt  A
) ) )
1614, 15eqeq12d 2452 . . 3  |-  ( x  =  -u y  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ -u y
) )  =  (
-u y  x.  ( P  pCnt  A ) ) ) )
17 oveq2 6092 . . . . 5  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
1817oveq2d 6100 . . . 4  |-  ( x  =  N  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ N ) ) )
19 oveq1 6091 . . . 4  |-  ( x  =  N  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( N  x.  ( P  pCnt  A ) ) )
2018, 19eqeq12d 2452 . . 3  |-  ( x  =  N  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ N
) )  =  ( N  x.  ( P 
pCnt  A ) ) ) )
21 pc1 13234 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2221adantr 453 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  1
)  =  0 )
23 qcn 10593 . . . . . . 7  |-  ( A  e.  QQ  ->  A  e.  CC )
2423ad2antrl 710 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  ->  A  e.  CC )
2524exp0d 11522 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( A ^ 0 )  =  1 )
2625oveq2d 6100 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( P  pCnt  1 ) )
27 pcqcl 13235 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  ZZ )
2827zcnd 10381 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  CC )
2928mul02d 9269 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( 0  x.  ( P  pCnt  A ) )  =  0 )
3022, 26, 293eqtr4d 2480 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P  pCnt  A
) ) )
31 oveq1 6091 . . . . 5  |-  ( ( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A
) )  ->  (
( P  pCnt  ( A ^ y ) )  +  ( P  pCnt  A ) )  =  ( ( y  x.  ( P  pCnt  A ) )  +  ( P  pCnt  A ) ) )
32 expp1 11393 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ (
y  +  1 ) )  =  ( ( A ^ y )  x.  A ) )
3324, 32sylan 459 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ ( y  +  1 ) )  =  ( ( A ^
y )  x.  A
) )
3433oveq2d 6100 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( P  pCnt  ( A ^
( y  +  1 ) ) )  =  ( P  pCnt  (
( A ^ y
)  x.  A ) ) )
35 simpll 732 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  P  e.  Prime )
36 simplrl 738 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  QQ )
37 simplrr 739 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  =/=  0 )
38 nn0z 10309 . . . . . . . . . 10  |-  ( y  e.  NN0  ->  y  e.  ZZ )
3938adantl 454 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  ZZ )
40 qexpclz 11407 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  A  =/=  0  /\  y  e.  ZZ )  ->  ( A ^ y )  e.  QQ )
4136, 37, 39, 40syl3anc 1185 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  e.  QQ )
4224adantr 453 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  CC )
4342, 37, 39expne0d 11534 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  =/=  0 )
44 pcqmul 13232 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
( A ^ y
)  e.  QQ  /\  ( A ^ y )  =/=  0 )  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  ->  ( P  pCnt  ( ( A ^
y )  x.  A
) )  =  ( ( P  pCnt  ( A ^ y ) )  +  ( P  pCnt  A ) ) )
4535, 41, 43, 36, 37, 44syl122anc 1194 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( P  pCnt  ( ( A ^ y )  x.  A ) )  =  ( ( P  pCnt  ( A ^ y ) )  +  ( P 
pCnt  A ) ) )
4634, 45eqtrd 2470 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( P  pCnt  ( A ^
( y  +  1 ) ) )  =  ( ( P  pCnt  ( A ^ y ) )  +  ( P 
pCnt  A ) ) )
47 nn0cn 10236 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
4847adantl 454 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  CC )
49 ax-1cn 9053 . . . . . . . . 9  |-  1  e.  CC
5049a1i 11 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  1  e.  CC )
5128adantr 453 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( P  pCnt  A )  e.  CC )
5248, 50, 51adddird 9118 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
( y  +  1 )  x.  ( P 
pCnt  A ) )  =  ( ( y  x.  ( P  pCnt  A
) )  +  ( 1  x.  ( P 
pCnt  A ) ) ) )
5351mulid2d 9111 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
1  x.  ( P 
pCnt  A ) )  =  ( P  pCnt  A
) )
5453oveq2d 6100 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
( y  x.  ( P  pCnt  A ) )  +  ( 1  x.  ( P  pCnt  A
) ) )  =  ( ( y  x.  ( P  pCnt  A
) )  +  ( P  pCnt  A )
) )
5552, 54eqtrd 2470 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
( y  +  1 )  x.  ( P 
pCnt  A ) )  =  ( ( y  x.  ( P  pCnt  A
) )  +  ( P  pCnt  A )
) )
5646, 55eqeq12d 2452 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
( P  pCnt  ( A ^ ( y  +  1 ) ) )  =  ( ( y  +  1 )  x.  ( P  pCnt  A
) )  <->  ( ( P  pCnt  ( A ^
y ) )  +  ( P  pCnt  A
) )  =  ( ( y  x.  ( P  pCnt  A ) )  +  ( P  pCnt  A ) ) ) )
5731, 56syl5ibr 214 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A
) )  ->  ( P  pCnt  ( A ^
( y  +  1 ) ) )  =  ( ( y  +  1 )  x.  ( P  pCnt  A ) ) ) )
5857ex 425 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( y  e.  NN0  ->  ( ( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A ) )  ->  ( P  pCnt  ( A ^
( y  +  1 ) ) )  =  ( ( y  +  1 )  x.  ( P  pCnt  A ) ) ) ) )
59 negeq 9303 . . . . 5  |-  ( ( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A
) )  ->  -u ( P  pCnt  ( A ^
y ) )  = 
-u ( y  x.  ( P  pCnt  A
) ) )
60 nnnn0 10233 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  NN0 )
61 expneg 11394 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ -u y
)  =  ( 1  /  ( A ^
y ) ) )
6224, 60, 61syl2an 465 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ -u y )  =  ( 1  / 
( A ^ y
) ) )
6362oveq2d 6100 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  =  ( P  pCnt  (
1  /  ( A ^ y ) ) ) )
64 simpll 732 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  P  e.  Prime )
6560, 41sylan2 462 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  e.  QQ )
6660, 43sylan2 462 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  =/=  0 )
67 pcrec 13237 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
( A ^ y
)  e.  QQ  /\  ( A ^ y )  =/=  0 ) )  ->  ( P  pCnt  ( 1  /  ( A ^ y ) ) )  =  -u ( P  pCnt  ( A ^
y ) ) )
6864, 65, 66, 67syl12anc 1183 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( 1  / 
( A ^ y
) ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
6963, 68eqtrd 2470 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
70 nncn 10013 . . . . . . 7  |-  ( y  e.  NN  ->  y  e.  CC )
71 mulneg1 9475 . . . . . . 7  |-  ( ( y  e.  CC  /\  ( P  pCnt  A )  e.  CC )  -> 
( -u y  x.  ( P  pCnt  A ) )  =  -u ( y  x.  ( P  pCnt  A
) ) )
7270, 28, 71syl2anr 466 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( -u y  x.  ( P 
pCnt  A ) )  = 
-u ( y  x.  ( P  pCnt  A
) ) )
7369, 72eqeq12d 2452 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  (
( P  pCnt  ( A ^ -u y ) )  =  ( -u y  x.  ( P  pCnt  A ) )  <->  -u ( P 
pCnt  ( A ^
y ) )  = 
-u ( y  x.  ( P  pCnt  A
) ) ) )
7459, 73syl5ibr 214 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  (
( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A
) )  ->  ( P  pCnt  ( A ^ -u y ) )  =  ( -u y  x.  ( P  pCnt  A
) ) ) )
7574ex 425 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( y  e.  NN  ->  ( ( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A ) )  ->  ( P  pCnt  ( A ^ -u y ) )  =  ( -u y  x.  ( P  pCnt  A
) ) ) ) )
764, 8, 12, 16, 20, 30, 58, 75zindd 10376 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( N  e.  ZZ  ->  ( P  pCnt  ( A ^ N ) )  =  ( N  x.  ( P  pCnt  A ) ) ) )
77763impia 1151 1  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  ( P  pCnt  ( A ^ N ) )  =  ( N  x.  ( P  pCnt  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000   -ucneg 9297    / cdiv 9682   NNcn 10005   NN0cn0 10226   ZZcz 10287   QQcq 10579   ^cexp 11387   Primecprime 13084    pCnt cpc 13215
This theorem is referenced by:  qexpz  13275  expnprm  13276  dchrisum0flblem1  21207  dchrisum0flblem2  21208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-dvds 12858  df-gcd 13012  df-prm 13085  df-pc 13216
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