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Theorem pcprmpw2 12950
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 731 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN )
21nnnn0d 10034 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  NN0 )
3 prmnn 12777 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
43ad2antrr 706 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  P  e.  NN )
5 pccl 12918 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P  pCnt  A )  e. 
NN0 )
65adantr 451 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  NN0 )
74, 6nnexpcld 11282 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
87nnnn0d 10034 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN0 )
96nn0red 10035 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  RR )
109leidd 9355 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  <_  ( P  pCnt  A ) )
11 simpll 730 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  P  e.  Prime )
126nn0zd 10131 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  e.  ZZ )
13 pcid 12941 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  ( P  pCnt  A )  e.  ZZ )  ->  ( P  pCnt  ( P ^
( P  pCnt  A
) ) )  =  ( P  pCnt  A
) )
1411, 12, 13syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  ( P ^ ( P  pCnt  A ) ) )  =  ( P 
pCnt  A ) )
1510, 14breqtrrd 4065 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P  pCnt  A )  <_  ( P  pCnt  ( P ^
( P  pCnt  A
) ) ) )
1615ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( ( 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 447 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  p  =  P )
1817oveq1d 5889 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =  P )  ->  ( p  pCnt  A )  =  ( P  pCnt  A )
)
1917oveq1d 5889 . . . . . . . . 9  |-  ( ( ( ( ( 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 4069 . . . . . . . 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 ) ) ) )
21 simplrr 737 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  ||  ( P ^ n ) )
22 prmz 12778 . . . . . . . . . . . . . . . 16  |-  ( p  e.  Prime  ->  p  e.  ZZ )
2322adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  p  e.  ZZ )
241adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  e.  NN )
2524nnzd 10132 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  A  e.  ZZ )
26 simprl 732 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  n  e.  NN0 )
274, 26nnexpcld 11282 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ n )  e.  NN )
2827adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  NN )
2928nnzd 10132 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( P ^ n )  e.  ZZ )
30 dvdstr 12578 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  ( P ^ n )  e.  ZZ )  ->  (
( p  ||  A  /\  A  ||  ( P ^ n ) )  ->  p  ||  ( P ^ n ) ) )
3123, 25, 29, 30syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 655 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  A  ->  p  ||  ( P ^ n
) ) )
33 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  p  e. 
Prime )
3411adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  P  e. 
Prime )
35 simplrl 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  n  e. 
NN0 )
36 prmdvdsexpr 12811 . . . . . . . . . . . . . 14  |-  ( ( p  e.  Prime  /\  P  e.  Prime  /\  n  e.  NN0 )  ->  ( p  ||  ( P ^ n
)  ->  p  =  P ) )
3733, 34, 35, 36syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  ( P ^
n )  ->  p  =  P ) )
3832, 37syld 40 . . . . . . . . . . . 12  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p 
||  A  ->  p  =  P ) )
3938necon3ad 2495 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  /\  p  e. 
Prime )  ->  ( p  =/=  P  ->  -.  p  ||  A ) )
4039imp 418 . . . . . . . . . 10  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  -.  p  ||  A )
41 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  p  e.  Prime )
421ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  A  e.  NN )
43 pceq0 12939 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  =  0  <->  -.  p  ||  A ) )
4441, 42, 43syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( ( 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 223 . . . . . . . . 9  |-  ( ( ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  (
n  e.  NN0  /\  A  ||  ( P ^
n ) ) )  /\  p  e.  Prime )  /\  p  =/=  P
)  ->  ( p  pCnt  A )  =  0 )
467ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 12919 . . . . . . . . . 10  |-  ( ( ( ( ( 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 10037 . . . . . . . . 9  |-  ( ( ( ( ( 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 4059 . . . . . . . 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 ) ) ) )
5020, 49pm2.61dane 2537 . . . . . . 7  |-  ( ( ( ( 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 2639 . . . . . 6  |-  ( ( ( 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 10132 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  e.  ZZ )
537nnzd 10132 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
54 pc2dvds 12947 . . . . . . 7  |-  ( ( 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 642 . . . . . 6  |-  ( ( ( 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 223 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  ||  ( P ^ ( P  pCnt  A ) ) )
57 pcdvds 12932 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
5857adantr 451 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
59 dvdseq 12592 . . . . 5  |-  ( ( ( 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 1183 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  ( n  e. 
NN0  /\  A  ||  ( P ^ n ) ) )  ->  A  =  ( P ^ ( P 
pCnt  A ) ) )
6160expr 598 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  NN )  /\  n  e.  NN0 )  ->  ( A  ||  ( P ^ n )  ->  A  =  ( P ^ ( P 
pCnt  A ) ) ) )
6261rexlimdva 2680 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  ->  A  =  ( P ^
( P  pCnt  A
) ) ) )
633adantr 451 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  P  e.  NN )
6463, 5nnexpcld 11282 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
6564nnzd 10132 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
66 iddvds 12558 . . . . 5  |-  ( ( P ^ ( P 
pCnt  A ) )  e.  ZZ  ->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) )
6765, 66syl 15 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) )
68 oveq2 5882 . . . . . 6  |-  ( n  =  ( P  pCnt  A )  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  A ) ) )
6968breq2d 4051 . . . . 5  |-  ( n  =  ( P  pCnt  A )  ->  ( ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ ( P  pCnt  A ) ) ) )
7069rspcev 2897 . . . 4  |-  ( ( ( P  pCnt  A
)  e.  NN0  /\  ( P ^ ( P 
pCnt  A ) )  ||  ( P ^ ( P 
pCnt  A ) ) )  ->  E. n  e.  NN0  ( P ^ ( P 
pCnt  A ) )  ||  ( P ^ n ) )
715, 67, 70syl2anc 642 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  E. n  e.  NN0  ( P ^
( P  pCnt  A
) )  ||  ( P ^ n ) )
72 breq1 4042 . . . 4  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( A  ||  ( P ^
n )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( P ^ n ) ) )
7372rexbidv 2577 . . 3  |-  ( A  =  ( P ^
( P  pCnt  A
) )  ->  ( E. n  e.  NN0  A 
||  ( P ^
n )  <->  E. n  e.  NN0  ( P ^
( P  pCnt  A
) )  ||  ( P ^ n ) ) )
7471, 73syl5ibrcom 213 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( A  =  ( P ^ ( P  pCnt  A ) )  ->  E. n  e.  NN0  A  ||  ( P ^ n ) ) )
7562, 74impbid 183 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   0cc0 8753    <_ cle 8884   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120    || cdivides 12547   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  pcprmpw  12951  pgpfi1  14922  pgpfi  14932  sylow2alem2  14945  lt6abl  15197  pgpfac1lem3a  15327  dvdsppwf1o  20442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906
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