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Theorem pcprmpw2 13210
Description: Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
pcprmpw2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  A  =  ( P ^ ( P 
pCnt  A ) ) ) )
Distinct variable groups:    A, n    P, n

Proof of Theorem pcprmpw2
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simplr 732 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN )
21nnnn0d 10230 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN0 )
3 prmnn 13037 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
43ad2antrr 707 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  P  e.  NN )
5 pccl 13178 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P  pCnt  A )  e. 
NN0 )
65adantr 452 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  NN0 )
74, 6nnexpcld 11499 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
87nnnn0d 10230 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN0 )
96nn0red 10231 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  RR )
109leidd 9549 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  <_  ( P  pCnt  A ) )
11 simpll 731 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  P  e.  Prime )
126nn0zd 10329 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  ZZ )
13 pcid 13201 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( P  pCnt  A )  e.  ZZ )  ->  ( P  pCnt  ( P ^
( P  pCnt  A
) ) )  =  ( P  pCnt  A
) )
1411, 12, 13syl2anc 643 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  ( P ^ ( P  pCnt  A ) ) )  =  ( P 
pCnt  A ) )
1510, 14breqtrrd 4198 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  <_  ( P  pCnt  ( P ^
( P  pCnt  A
) ) ) )
1615ad2antrr 707 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( P  pCnt  A )  <_  ( P  pCnt  ( P ^
( P  pCnt  A
) ) ) )
17 simpr 448 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  p  =  P )
1817oveq1d 6055 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( p  pCnt  A )  =  ( P  pCnt  A )
)
1917oveq1d 6055 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) )  =  ( P 
pCnt  ( P ^
( P  pCnt  A
) ) ) )
2016, 18, 193brtr4d 4202 . . . . . . 7  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( p  pCnt  A )  <_  (
p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
21 simplrr 738 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  ||  ( P ^ n ) )
22 prmz 13038 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
2322adantl 453 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  p  e.  ZZ )
241adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  e.  NN )
2524nnzd 10330 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  e.  ZZ )
26 simprl 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  n  e.  NN0 )
274, 26nnexpcld 11499 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ n )  e.  NN )
2827adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  NN )
2928nnzd 10330 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  ZZ )
30 dvdstr 12838 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  ( P ^ n )  e.  ZZ )  ->  (
( p  ||  A  /\  A  ||  ( P ^ n ) )  ->  p  ||  ( P ^ n ) ) )
3123, 25, 29, 30syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( ( p  ||  A  /\  A  ||  ( P ^
n ) )  ->  p  ||  ( P ^
n ) ) )
3221, 31mpan2d 656 . . . . . . . . . . . 12  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  A  ->  p  ||  ( P ^ n
) ) )
33 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  p  e. 
Prime )
3411adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  P  e. 
Prime )
35 simplrl 737 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  n  e. 
NN0 )
36 prmdvdsexpr 13071 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  P  e.  Prime  /\  n  e.  NN0 )  ->  ( p  ||  ( P ^ n
)  ->  p  =  P ) )
3733, 34, 35, 36syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  ( P ^
n )  ->  p  =  P ) )
3832, 37syld 42 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  A  ->  p  =  P ) )
3938necon3ad 2603 . . . . . . . . . 10  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p  =/=  P  ->  -.  p  ||  A ) )
4039imp 419 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  -.  p  ||  A )
41 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  p  e.  Prime )
421ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  A  e.  NN )
43 pceq0 13199 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  =  0  <->  -.  p  ||  A ) )
4441, 42, 43syl2anc 643 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( (
p  pCnt  A )  =  0  <->  -.  p  ||  A ) )
4540, 44mpbird 224 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( p  pCnt  A )  =  0 )
467ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
4741, 46pccld 13179 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) )  e.  NN0 )
4847nn0ge0d 10233 . . . . . . . 8  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  0  <_  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
4945, 48eqbrtrd 4192 . . . . . . 7  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( p  pCnt  A )  <_  (
p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
5020, 49pm2.61dane 2645 . . . . . 6  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
pCnt  A )  <_  (
p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
5150ralrimiva 2749 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A. p  e.  Prime  ( p  pCnt  A )  <_  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) ) )
521nnzd 10330 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  ZZ )
537nnzd 10330 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
54 pc2dvds 13207 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( P ^ ( P 
pCnt  A ) )  e.  ZZ )  ->  ( A  ||  ( P ^
( P  pCnt  A
) )  <->  A. p  e.  Prime  ( p  pCnt  A )  <_  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) ) ) )
5552, 53, 54syl2anc 643 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( A  ||  ( P ^ ( P  pCnt  A ) )  <->  A. p  e.  Prime  ( p  pCnt  A )  <_  ( p  pCnt  ( P ^ ( P  pCnt  A ) ) ) ) )
5651, 55mpbird 224 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  ||  ( P ^ ( P  pCnt  A ) ) )
57 pcdvds 13192 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
5857adantr 452 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
59 dvdseq 12852 . . . 4  |-  ( ( ( A  e.  NN0  /\  ( P ^ ( P  pCnt  A ) )  e.  NN0 )  /\  ( A  ||  ( P ^ ( P  pCnt  A ) )  /\  ( P ^ ( P  pCnt  A ) )  ||  A
) )  ->  A  =  ( P ^
( P  pCnt  A
) ) )
602, 8, 56, 58, 59syl22anc 1185 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  =  ( P ^ ( P 
pCnt  A ) ) )
6160rexlimdvaa 2791 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  ->  A  =  ( P ^
( P  pCnt  A
) ) ) )
623adantr 452 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  P  e.  NN )
6362, 5nnexpcld 11499 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
6463nnzd 10330 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
65 iddvds 12818 . . . . 5  |-  ( ( P ^ ( P 
pCnt  A ) )  e.  ZZ  ->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) )
6664, 65syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) )
67 oveq2 6048 . . . . . 6  |-  ( n  =  ( P  pCnt  A )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  A ) ) )
6867breq2d 4184 . . . . 5  |-  ( n  =  ( P  pCnt  A )  ->  ( ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) ) )
6968rspcev 3012 . . . 4  |-  ( ( ( P  pCnt  A
)  e.  NN0  /\  ( P ^ ( P 
pCnt  A ) )  ||  ( P ^ ( P 
pCnt  A ) ) )  ->  E. n  e.  NN0  ( P ^ ( P 
pCnt  A ) )  ||  ( P ^ n ) )
705, 66, 69syl2anc 643 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  E. n  e.  NN0  ( P ^
( P  pCnt  A
) )  ||  ( P ^ n ) )
71 breq1 4175 . . . 4  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( A  ||  ( P ^
n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n ) ) )
7271rexbidv 2687 . . 3  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  E. n  e.  NN0  ( P ^
( P  pCnt  A
) )  ||  ( P ^ n ) ) )
7370, 72syl5ibrcom 214 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( A  =  ( P ^ ( P  pCnt  A ) )  ->  E. n  e.  NN0  A  ||  ( P ^ n ) ) )
7461, 73impbid 184 1  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  A  =  ( P ^ ( P 
pCnt  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   class class class wbr 4172  (class class class)co 6040   0cc0 8946    <_ cle 9077   NNcn 9956   NN0cn0 10177   ZZcz 10238   ^cexp 11337    || cdivides 12807   Primecprime 13034    pCnt cpc 13165
This theorem is referenced by:  pcprmpw  13211  pgpfi1  15184  pgpfi  15194  sylow2alem2  15207  lt6abl  15459  pgpfac1lem3a  15589  dvdsppwf1o  20924
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166
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