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Theorem pcexp 13929
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 6102 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
21oveq2d 6110 . . . 4  |-  ( x  =  0  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ 0 ) ) )
3 oveq1 6101 . . . 4  |-  ( x  =  0  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( 0  x.  ( P  pCnt  A ) ) )
42, 3eqeq12d 2457 . . 3  |-  ( x  =  0  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P 
pCnt  A ) ) ) )
5 oveq2 6102 . . . . 5  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
65oveq2d 6110 . . . 4  |-  ( x  =  y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ y ) ) )
7 oveq1 6101 . . . 4  |-  ( x  =  y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( y  x.  ( P  pCnt  A ) ) )
86, 7eqeq12d 2457 . . 3  |-  ( x  =  y  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ y
) )  =  ( y  x.  ( P 
pCnt  A ) ) ) )
9 oveq2 6102 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A ^ x )  =  ( A ^ (
y  +  1 ) ) )
109oveq2d 6110 . . . 4  |-  ( x  =  ( y  +  1 )  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ ( y  +  1 ) ) ) )
11 oveq1 6101 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( ( y  +  1 )  x.  ( P  pCnt  A ) ) )
1210, 11eqeq12d 2457 . . 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 6102 . . . . 5  |-  ( x  =  -u y  ->  ( A ^ x )  =  ( A ^ -u y
) )
1413oveq2d 6110 . . . 4  |-  ( x  =  -u y  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ -u y ) ) )
15 oveq1 6101 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( -u y  x.  ( P  pCnt  A
) ) )
1614, 15eqeq12d 2457 . . 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 6102 . . . . 5  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
1817oveq2d 6110 . . . 4  |-  ( x  =  N  ->  ( P  pCnt  ( A ^
x ) )  =  ( P  pCnt  ( A ^ N ) ) )
19 oveq1 6101 . . . 4  |-  ( x  =  N  ->  (
x  x.  ( P 
pCnt  A ) )  =  ( N  x.  ( P  pCnt  A ) ) )
2018, 19eqeq12d 2457 . . 3  |-  ( x  =  N  ->  (
( P  pCnt  ( A ^ x ) )  =  ( x  x.  ( P  pCnt  A
) )  <->  ( P  pCnt  ( A ^ N
) )  =  ( N  x.  ( P 
pCnt  A ) ) ) )
21 pc1 13925 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2221adantr 465 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  1
)  =  0 )
23 qcn 10970 . . . . . . 7  |-  ( A  e.  QQ  ->  A  e.  CC )
2423ad2antrl 727 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  ->  A  e.  CC )
2524exp0d 12005 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( A ^ 0 )  =  1 )
2625oveq2d 6110 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( P  pCnt  1 ) )
27 pcqcl 13926 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  ZZ )
2827zcnd 10751 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  CC )
2928mul02d 9570 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( 0  x.  ( P  pCnt  A ) )  =  0 )
3022, 26, 293eqtr4d 2485 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( P  pCnt  ( A ^ 0 ) )  =  ( 0  x.  ( P  pCnt  A
) ) )
31 oveq1 6101 . . . . 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 11875 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ (
y  +  1 ) )  =  ( ( A ^ y )  x.  A ) )
3324, 32sylan 471 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ ( y  +  1 ) )  =  ( ( A ^
y )  x.  A
) )
3433oveq2d 6110 . . . . . . 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 753 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  P  e.  Prime )
36 simplrl 759 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  QQ )
37 simplrr 760 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  =/=  0 )
38 nn0z 10672 . . . . . . . . . 10  |-  ( y  e.  NN0  ->  y  e.  ZZ )
3938adantl 466 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  ZZ )
40 qexpclz 11889 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  A  =/=  0  /\  y  e.  ZZ )  ->  ( A ^ y )  e.  QQ )
4136, 37, 39, 40syl3anc 1218 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  e.  QQ )
4224adantr 465 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  A  e.  CC )
4342, 37, 39expne0d 12017 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( A ^ y )  =/=  0 )
44 pcqmul 13923 . . . . . . . 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 1227 . . . . . . 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 2475 . . . . . 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 10592 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
4847adantl 466 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  y  e.  CC )
49 ax-1cn 9343 . . . . . . . . 9  |-  1  e.  CC
5049a1i 11 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  1  e.  CC )
5128adantr 465 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  ( P  pCnt  A )  e.  CC )
5248, 50, 51adddird 9414 . . . . . . 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 9407 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
1  x.  ( P 
pCnt  A ) )  =  ( P  pCnt  A
) )
5453oveq2d 6110 . . . . . . 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 2475 . . . . . 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 2457 . . . . 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 9605 . . . . 5  |-  ( ( P  pCnt  ( A ^ y ) )  =  ( y  x.  ( P  pCnt  A
) )  ->  -u ( P  pCnt  ( A ^
y ) )  = 
-u ( y  x.  ( P  pCnt  A
) ) )
60 nnnn0 10589 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  NN0 )
61 expneg 11876 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ -u y
)  =  ( 1  /  ( A ^
y ) ) )
6224, 60, 61syl2an 477 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ -u y )  =  ( 1  / 
( A ^ y
) ) )
6362oveq2d 6110 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  =  ( P  pCnt  (
1  /  ( A ^ y ) ) ) )
64 simpll 753 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  P  e.  Prime )
6560, 41sylan2 474 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  e.  QQ )
6660, 43sylan2 474 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( A ^ y )  =/=  0 )
67 pcrec 13928 . . . . . . . 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 1216 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( 1  / 
( A ^ y
) ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
6963, 68eqtrd 2475 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  /\  y  e.  NN )  ->  ( P  pCnt  ( A ^ -u y ) )  = 
-u ( P  pCnt  ( A ^ y ) ) )
70 nncn 10333 . . . . . . 7  |-  ( y  e.  NN  ->  y  e.  CC )
71 mulneg1 9784 . . . . . . 7  |-  ( ( y  e.  CC  /\  ( P  pCnt  A )  e.  CC )  -> 
( -u y  x.  ( P  pCnt  A ) )  =  -u ( y  x.  ( P  pCnt  A
) ) )
7270, 28, 71syl2anr 478 . . . . . 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 2457 . . . . 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 10746 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( N  e.  ZZ  ->  ( P  pCnt  ( A ^ N ) )  =  ( N  x.  ( P  pCnt  A ) ) ) )
77763impia 1184 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 965    = wceq 1369    e. wcel 1756    =/= wne 2609  (class class class)co 6094   CCcc 9283   0cc0 9285   1c1 9286    + caddc 9288    x. cmul 9290   -ucneg 9599    / cdiv 9996   NNcn 10325   NN0cn0 10582   ZZcz 10649   QQcq 10956   ^cexp 11868   Primecprime 13766    pCnt cpc 13906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-dvds 13539  df-gcd 13694  df-prm 13767  df-pc 13907
This theorem is referenced by:  qexpz  13966  expnprm  13967  dchrisum0flblem1  22760  dchrisum0flblem2  22761
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