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Theorem pcexp 12912
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 5866 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
21oveq2d 5874 . . . 4  |-  ( x  =  0  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ 0 ) ) )
3 oveq1 5865 . . . 4  |-  ( x  =  0  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( 0  x.  ( P  pCnt  A ) ) )
42, 3eqeq12d 2297 . . 3  |-  ( x  =  0  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P 
pCnt  A ) ) ) )
5 oveq2 5866 . . . . 5  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
65oveq2d 5874 . . . 4  |-  ( x  =  y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ y ) ) )
7 oveq1 5865 . . . 4  |-  ( x  =  y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( y  x.  ( P  pCnt  A ) ) )
86, 7eqeq12d 2297 . . 3  |-  ( x  =  y  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ y
) )  =  ( y  x.  ( P 
pCnt  A ) ) ) )
9 oveq2 5866 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A ^ x )  =  ( A ^ (
y  +  1 ) ) )
109oveq2d 5874 . . . 4  |-  ( x  =  ( y  +  1 )  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ ( y  +  1 ) ) ) )
11 oveq1 5865 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( ( y  +  1 )  x.  ( P  pCnt  A ) ) )
1210, 11eqeq12d 2297 . . 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 5866 . . . . 5  |-  ( x  =  -u y  ->  ( A ^ x )  =  ( A ^ -u y
) )
1413oveq2d 5874 . . . 4  |-  ( x  =  -u y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ -u y ) ) )
15 oveq1 5865 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( -u y  x.  ( P  pCnt  A
) ) )
1614, 15eqeq12d 2297 . . 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 5866 . . . . 5  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
1817oveq2d 5874 . . . 4  |-  ( x  =  N  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ N ) ) )
19 oveq1 5865 . . . 4  |-  ( x  =  N  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( N  x.  ( P  pCnt  A ) ) )
2018, 19eqeq12d 2297 . . 3  |-  ( x  =  N  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ N
) )  =  ( N  x.  ( P 
pCnt  A ) ) ) )
21 pc1 12908 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2221adantr 451 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  1
)  =  0 )
23 qcn 10330 . . . . . . 7  |-  ( A  e.  QQ  ->  A  e.  CC )
2423ad2antrl 708 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  ->  A  e.  CC )
2524exp0d 11239 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( A ^ 0 )  =  1 )
2625oveq2d 5874 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( P  pCnt  1 ) )
27 pcqcl 12909 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  ZZ )
2827zcnd 10118 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  CC )
2928mul02d 9010 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( 0  x.  ( P  pCnt  A ) )  =  0 )
3022, 26, 293eqtr4d 2325 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P  pCnt  A
) ) )
31 oveq1 5865 . . . . 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 11110 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ (
y  +  1 ) )  =  ( ( A ^ y )  x.  A ) )
3324, 32sylan 457 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ ( y  +  1 ) )  =  ( ( A ^
y )  x.  A
) )
3433oveq2d 5874 . . . . . . 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 730 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  P  e.  Prime )
36 simplrl 736 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  QQ )
37 simplrr 737 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  =/=  0 )
38 nn0z 10046 . . . . . . . . . 10  |-  ( y  e.  NN0  ->  y  e.  ZZ )
3938adantl 452 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  ZZ )
40 qexpclz 11124 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  A  =/=  0  /\  y  e.  ZZ )  ->  ( A ^ y )  e.  QQ )
4136, 37, 39, 40syl3anc 1182 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  e.  QQ )
4224adantr 451 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  CC )
4342, 37, 39expne0d 11251 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  =/=  0 )
44 pcqmul 12906 . . . . . . . 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 1191 . . . . . . 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 2315 . . . . . 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 9975 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
4847adantl 452 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  CC )
49 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
5049a1i 10 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  1  e.  CC )
5128adantr 451 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( P  pCnt  A )  e.  CC )
5248, 50, 51adddird 8860 . . . . . . 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 8853 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
1  x.  ( P 
pCnt  A ) )  =  ( P  pCnt  A
) )
5453oveq2d 5874 . . . . . . 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 2315 . . . . . 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 2297 . . . . 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 212 . . . 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 423 . . 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 9044 . . . . 5  |-  ( ( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A
) )  ->  -u ( P  pCnt  ( A ^
y ) )  = 
-u ( y  x.  ( P  pCnt  A
) ) )
60 nnnn0 9972 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  NN0 )
61 expneg 11111 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ -u y
)  =  ( 1  /  ( A ^
y ) ) )
6224, 60, 61syl2an 463 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ -u y )  =  ( 1  / 
( A ^ y
) ) )
6362oveq2d 5874 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  =  ( P  pCnt  (
1  /  ( A ^ y ) ) ) )
64 simpll 730 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  P  e.  Prime )
6560, 41sylan2 460 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  e.  QQ )
6660, 43sylan2 460 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  =/=  0 )
67 pcrec 12911 . . . . . . . 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 1180 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( 1  / 
( A ^ y
) ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
6963, 68eqtrd 2315 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
70 nncn 9754 . . . . . . 7  |-  ( y  e.  NN  ->  y  e.  CC )
71 mulneg1 9216 . . . . . . 7  |-  ( ( y  e.  CC  /\  ( P  pCnt  A )  e.  CC )  -> 
( -u y  x.  ( P  pCnt  A ) )  =  -u ( y  x.  ( P  pCnt  A
) ) )
7270, 28, 71syl2anr 464 . . . . . 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 2297 . . . . 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 212 . . . 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 423 . . 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 10113 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( N  e.  ZZ  ->  ( P  pCnt  ( A ^ N ) )  =  ( N  x.  ( P  pCnt  A ) ) ) )
77763impia 1148 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   QQcq 10316   ^cexp 11104   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  qexpz  12949  expnprm  12950  dchrisum0flblem1  20657  dchrisum0flblem2  20658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890
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