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Theorem pcexp 13196
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 6056 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
21oveq2d 6064 . . . 4  |-  ( x  =  0  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ 0 ) ) )
3 oveq1 6055 . . . 4  |-  ( x  =  0  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( 0  x.  ( P  pCnt  A ) ) )
42, 3eqeq12d 2426 . . 3  |-  ( x  =  0  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P 
pCnt  A ) ) ) )
5 oveq2 6056 . . . . 5  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
65oveq2d 6064 . . . 4  |-  ( x  =  y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ y ) ) )
7 oveq1 6055 . . . 4  |-  ( x  =  y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( y  x.  ( P  pCnt  A ) ) )
86, 7eqeq12d 2426 . . 3  |-  ( x  =  y  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ y
) )  =  ( y  x.  ( P 
pCnt  A ) ) ) )
9 oveq2 6056 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A ^ x )  =  ( A ^ (
y  +  1 ) ) )
109oveq2d 6064 . . . 4  |-  ( x  =  ( y  +  1 )  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ ( y  +  1 ) ) ) )
11 oveq1 6055 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( ( y  +  1 )  x.  ( P  pCnt  A ) ) )
1210, 11eqeq12d 2426 . . 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 6056 . . . . 5  |-  ( x  =  -u y  ->  ( A ^ x )  =  ( A ^ -u y
) )
1413oveq2d 6064 . . . 4  |-  ( x  =  -u y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ -u y ) ) )
15 oveq1 6055 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( -u y  x.  ( P  pCnt  A
) ) )
1614, 15eqeq12d 2426 . . 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 6056 . . . . 5  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
1817oveq2d 6064 . . . 4  |-  ( x  =  N  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ N ) ) )
19 oveq1 6055 . . . 4  |-  ( x  =  N  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( N  x.  ( P  pCnt  A ) ) )
2018, 19eqeq12d 2426 . . 3  |-  ( x  =  N  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ N
) )  =  ( N  x.  ( P 
pCnt  A ) ) ) )
21 pc1 13192 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2221adantr 452 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  1
)  =  0 )
23 qcn 10552 . . . . . . 7  |-  ( A  e.  QQ  ->  A  e.  CC )
2423ad2antrl 709 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  ->  A  e.  CC )
2524exp0d 11480 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( A ^ 0 )  =  1 )
2625oveq2d 6064 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( P  pCnt  1 ) )
27 pcqcl 13193 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  ZZ )
2827zcnd 10340 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  CC )
2928mul02d 9228 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( 0  x.  ( P  pCnt  A ) )  =  0 )
3022, 26, 293eqtr4d 2454 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P  pCnt  A
) ) )
31 oveq1 6055 . . . . 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 11351 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ (
y  +  1 ) )  =  ( ( A ^ y )  x.  A ) )
3324, 32sylan 458 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ ( y  +  1 ) )  =  ( ( A ^
y )  x.  A
) )
3433oveq2d 6064 . . . . . . 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 731 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  P  e.  Prime )
36 simplrl 737 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  QQ )
37 simplrr 738 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  =/=  0 )
38 nn0z 10268 . . . . . . . . . 10  |-  ( y  e.  NN0  ->  y  e.  ZZ )
3938adantl 453 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  ZZ )
40 qexpclz 11365 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  A  =/=  0  /\  y  e.  ZZ )  ->  ( A ^ y )  e.  QQ )
4136, 37, 39, 40syl3anc 1184 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  e.  QQ )
4224adantr 452 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  CC )
4342, 37, 39expne0d 11492 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  =/=  0 )
44 pcqmul 13190 . . . . . . . 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 1193 . . . . . . 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 2444 . . . . . 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 10195 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
4847adantl 453 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  CC )
49 ax-1cn 9012 . . . . . . . . 9  |-  1  e.  CC
5049a1i 11 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  1  e.  CC )
5128adantr 452 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( P  pCnt  A )  e.  CC )
5248, 50, 51adddird 9077 . . . . . . 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 9070 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
1  x.  ( P 
pCnt  A ) )  =  ( P  pCnt  A
) )
5453oveq2d 6064 . . . . . . 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 2444 . . . . . 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 2426 . . . . 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 213 . . . 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 424 . . 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 9262 . . . . 5  |-  ( ( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A
) )  ->  -u ( P  pCnt  ( A ^
y ) )  = 
-u ( y  x.  ( P  pCnt  A
) ) )
60 nnnn0 10192 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  NN0 )
61 expneg 11352 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ -u y
)  =  ( 1  /  ( A ^
y ) ) )
6224, 60, 61syl2an 464 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ -u y )  =  ( 1  / 
( A ^ y
) ) )
6362oveq2d 6064 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  =  ( P  pCnt  (
1  /  ( A ^ y ) ) ) )
64 simpll 731 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  P  e.  Prime )
6560, 41sylan2 461 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  e.  QQ )
6660, 43sylan2 461 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  =/=  0 )
67 pcrec 13195 . . . . . . . 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 1182 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( 1  / 
( A ^ y
) ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
6963, 68eqtrd 2444 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
70 nncn 9972 . . . . . . 7  |-  ( y  e.  NN  ->  y  e.  CC )
71 mulneg1 9434 . . . . . . 7  |-  ( ( y  e.  CC  /\  ( P  pCnt  A )  e.  CC )  -> 
( -u y  x.  ( P  pCnt  A ) )  =  -u ( y  x.  ( P  pCnt  A
) ) )
7270, 28, 71syl2anr 465 . . . . . 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 2426 . . . . 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 213 . . . 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 424 . . 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 10335 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( N  e.  ZZ  ->  ( P  pCnt  ( A ^ N ) )  =  ( N  x.  ( P  pCnt  A ) ) ) )
77763impia 1150 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575  (class class class)co 6048   CCcc 8952   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959   -ucneg 9256    / cdiv 9641   NNcn 9964   NN0cn0 10185   ZZcz 10246   QQcq 10538   ^cexp 11345   Primecprime 13042    pCnt cpc 13173
This theorem is referenced by:  qexpz  13233  expnprm  13234  dchrisum0flblem1  21163  dchrisum0flblem2  21164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-dvds 12816  df-gcd 12970  df-prm 13043  df-pc 13174
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