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Theorem pcexp 13922
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 6098 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
21oveq2d 6106 . . . 4  |-  ( x  =  0  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ 0 ) ) )
3 oveq1 6097 . . . 4  |-  ( x  =  0  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( 0  x.  ( P  pCnt  A ) ) )
42, 3eqeq12d 2455 . . 3  |-  ( x  =  0  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P 
pCnt  A ) ) ) )
5 oveq2 6098 . . . . 5  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
65oveq2d 6106 . . . 4  |-  ( x  =  y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ y ) ) )
7 oveq1 6097 . . . 4  |-  ( x  =  y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( y  x.  ( P  pCnt  A ) ) )
86, 7eqeq12d 2455 . . 3  |-  ( x  =  y  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ y
) )  =  ( y  x.  ( P 
pCnt  A ) ) ) )
9 oveq2 6098 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A ^ x )  =  ( A ^ (
y  +  1 ) ) )
109oveq2d 6106 . . . 4  |-  ( x  =  ( y  +  1 )  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ ( y  +  1 ) ) ) )
11 oveq1 6097 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( ( y  +  1 )  x.  ( P  pCnt  A ) ) )
1210, 11eqeq12d 2455 . . 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 6098 . . . . 5  |-  ( x  =  -u y  ->  ( A ^ x )  =  ( A ^ -u y
) )
1413oveq2d 6106 . . . 4  |-  ( x  =  -u y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ -u y ) ) )
15 oveq1 6097 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( -u y  x.  ( P  pCnt  A
) ) )
1614, 15eqeq12d 2455 . . 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 6098 . . . . 5  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
1817oveq2d 6106 . . . 4  |-  ( x  =  N  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ N ) ) )
19 oveq1 6097 . . . 4  |-  ( x  =  N  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( N  x.  ( P  pCnt  A ) ) )
2018, 19eqeq12d 2455 . . 3  |-  ( x  =  N  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ N
) )  =  ( N  x.  ( P 
pCnt  A ) ) ) )
21 pc1 13918 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2221adantr 462 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  1
)  =  0 )
23 qcn 10963 . . . . . . 7  |-  ( A  e.  QQ  ->  A  e.  CC )
2423ad2antrl 722 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  ->  A  e.  CC )
2524exp0d 11998 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( A ^ 0 )  =  1 )
2625oveq2d 6106 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( P  pCnt  1 ) )
27 pcqcl 13919 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  ZZ )
2827zcnd 10744 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  CC )
2928mul02d 9563 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( 0  x.  ( P  pCnt  A ) )  =  0 )
3022, 26, 293eqtr4d 2483 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P  pCnt  A
) ) )
31 oveq1 6097 . . . . 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 11868 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ (
y  +  1 ) )  =  ( ( A ^ y )  x.  A ) )
3324, 32sylan 468 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ ( y  +  1 ) )  =  ( ( A ^
y )  x.  A
) )
3433oveq2d 6106 . . . . . . 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 748 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  P  e.  Prime )
36 simplrl 754 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  QQ )
37 simplrr 755 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  =/=  0 )
38 nn0z 10665 . . . . . . . . . 10  |-  ( y  e.  NN0  ->  y  e.  ZZ )
3938adantl 463 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  ZZ )
40 qexpclz 11882 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  A  =/=  0  /\  y  e.  ZZ )  ->  ( A ^ y )  e.  QQ )
4136, 37, 39, 40syl3anc 1213 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  e.  QQ )
4224adantr 462 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  CC )
4342, 37, 39expne0d 12010 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  =/=  0 )
44 pcqmul 13916 . . . . . . . 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 1222 . . . . . . 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 2473 . . . . . 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 10585 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
4847adantl 463 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  CC )
49 ax-1cn 9336 . . . . . . . . 9  |-  1  e.  CC
5049a1i 11 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  1  e.  CC )
5128adantr 462 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( P  pCnt  A )  e.  CC )
5248, 50, 51adddird 9407 . . . . . . 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 9400 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
1  x.  ( P 
pCnt  A ) )  =  ( P  pCnt  A
) )
5453oveq2d 6106 . . . . . . 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 2473 . . . . . 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 2455 . . . . 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 221 . . . 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 434 . . 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 9598 . . . . 5  |-  ( ( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A
) )  ->  -u ( P  pCnt  ( A ^
y ) )  = 
-u ( y  x.  ( P  pCnt  A
) ) )
60 nnnn0 10582 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  NN0 )
61 expneg 11869 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ -u y
)  =  ( 1  /  ( A ^
y ) ) )
6224, 60, 61syl2an 474 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ -u y )  =  ( 1  / 
( A ^ y
) ) )
6362oveq2d 6106 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  =  ( P  pCnt  (
1  /  ( A ^ y ) ) ) )
64 simpll 748 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  P  e.  Prime )
6560, 41sylan2 471 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  e.  QQ )
6660, 43sylan2 471 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  =/=  0 )
67 pcrec 13921 . . . . . . . 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 1211 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( 1  / 
( A ^ y
) ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
6963, 68eqtrd 2473 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
70 nncn 10326 . . . . . . 7  |-  ( y  e.  NN  ->  y  e.  CC )
71 mulneg1 9777 . . . . . . 7  |-  ( ( y  e.  CC  /\  ( P  pCnt  A )  e.  CC )  -> 
( -u y  x.  ( P  pCnt  A ) )  =  -u ( y  x.  ( P  pCnt  A
) ) )
7270, 28, 71syl2anr 475 . . . . . 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 2455 . . . . 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 221 . . . 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 434 . . 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 10739 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( N  e.  ZZ  ->  ( P  pCnt  ( A ^ N ) )  =  ( N  x.  ( P  pCnt  A ) ) ) )
77763impia 1179 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283   -ucneg 9592    / cdiv 9989   NNcn 10318   NN0cn0 10575   ZZcz 10642   QQcq 10949   ^cexp 11861   Primecprime 13759    pCnt cpc 13899
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-int 4126  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-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  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-q 10950  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  df-prm 13760  df-pc 13900
This theorem is referenced by:  qexpz  13959  expnprm  13960  dchrisum0flblem1  22716  dchrisum0flblem2  22717
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