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Theorem pcprmpw2 13247
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 10266 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN0 )
3 prmnn 13074 . . . . . . 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 13215 . . . . . . 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 11536 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
87nnnn0d 10266 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN0 )
96nn0red 10267 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  RR )
109leidd 9585 . . . . . . . . . 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 10365 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  ZZ )
13 pcid 13238 . . . . . . . . . . 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 4230 . . . . . . . . 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 6088 . . . . . . . 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 6088 . . . . . . . 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 4234 . . . . . . 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 13075 . . . . . . . . . . . . . . 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 10366 . . . . . . . . . . . . . 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 11536 . . . . . . . . . . . . . . . 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 10366 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  ZZ )
30 dvdstr 12875 . . . . . . . . . . . . . 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 13108 . . . . . . . . . . . . 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 2634 . . . . . . . . . 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 13236 . . . . . . . . . 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 13216 . . . . . . . . 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 10269 . . . . . . . 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 4224 . . . . . . 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 2676 . . . . . 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 2781 . . . . 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 10366 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  ZZ )
537nnzd 10366 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
54 pc2dvds 13244 . . . . . 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 13229 . . . . 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 12889 . . . 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 2823 . 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 11536 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
6463nnzd 10366 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
65 iddvds 12855 . . . . 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 6081 . . . . . 6  |-  ( n  =  ( P  pCnt  A )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  A ) ) )
6867breq2d 4216 . . . . 5  |-  ( n  =  ( P  pCnt  A )  ->  ( ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) ) )
6968rspcev 3044 . . . 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 4207 . . . 4  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( A  ||  ( P ^
n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n ) ) )
7271rexbidv 2718 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   class class class wbr 4204  (class class class)co 6073   0cc0 8982    <_ cle 9113   NNcn 9992   NN0cn0 10213   ZZcz 10274   ^cexp 11374    || cdivides 12844   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  pcprmpw  13248  pgpfi1  15221  pgpfi  15231  sylow2alem2  15244  lt6abl  15496  pgpfac1lem3a  15626  dvdsppwf1o  20963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203
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