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Theorem isppw2 20353
Description: Two ways to say that  A is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
isppw2  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^
k ) ) )
Distinct variable group:    k, p, A

Proof of Theorem isppw2
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 isppw 20352 . 2  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E! q  e.  Prime  q 
||  A ) )
2 reu6 2954 . . 3  |-  ( E! q  e.  Prime  q  ||  A  <->  E. p  e.  Prime  A. q  e.  Prime  (
q  ||  A  <->  q  =  p ) )
3 equid 1644 . . . . . . . . 9  |-  p  =  p
4 breq1 4026 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  ||  A  <->  p  ||  A
) )
5 equequ1 1648 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
q  =  p  <->  p  =  p ) )
64, 5bibi12d 312 . . . . . . . . . . 11  |-  ( q  =  p  ->  (
( q  ||  A  <->  q  =  p )  <->  ( p  ||  A  <->  p  =  p
) ) )
76rspcva 2882 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A. q  e.  Prime  ( q 
||  A  <->  q  =  p ) )  -> 
( p  ||  A  <->  p  =  p ) )
87adantll 694 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p  ||  A  <->  p  =  p ) )
93, 8mpbiri 224 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  p  ||  A )
10 simplr 731 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  p  e.  Prime )
11 simpll 730 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  e.  NN )
12 pcelnn 12922 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
1310, 11, 12syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( ( p  pCnt  A )  e.  NN  <->  p  ||  A
) )
149, 13mpbird 223 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p  pCnt  A
)  e.  NN )
15 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  q  =  p )
1615oveq1d 5873 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( p  pCnt  A )
)
17 simpllr 735 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  p  e.  Prime )
18 pccl 12902 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
1918ancoms 439 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  A
)  e.  NN0 )
2019ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  NN0 )
2120nn0zd 10115 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  A )  e.  ZZ )
22 pcid 12925 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  Prime  /\  (
p  pCnt  A )  e.  ZZ )  ->  (
p  pCnt  ( p ^ ( p  pCnt  A ) ) )  =  ( p  pCnt  A
) )
2317, 21, 22syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( p  pCnt  ( p ^ (
p  pCnt  A )
) )  =  ( p  pCnt  A )
)
2416, 23eqtr4d 2318 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( p  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
2515oveq1d 5873 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  ( p ^ (
p  pCnt  A )
) )  =  ( p  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
2624, 25eqtr4d 2318 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  q  =  p )  ->  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
27 simprr 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  ||  A  <->  q  =  p ) )
2827notbid 285 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( -.  q  ||  A  <->  -.  q  =  p ) )
2928biimpar 471 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  -.  q  ||  A )
30 simplrl 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  q  e.  Prime )
31 simplll 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  A  e.  NN )
32 pceq0 12923 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Prime  /\  A  e.  NN )  ->  (
( q  pCnt  A
)  =  0  <->  -.  q  ||  A ) )
3330, 31, 32syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
( q  pCnt  A
)  =  0  <->  -.  q  ||  A ) )
3429, 33mpbird 223 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
q  pCnt  A )  =  0 )
35 simprl 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  q  e.  Prime )
36 simplr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  p  e.  Prime )
3719adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( p  pCnt  A )  e.  NN0 )
38 prmdvdsexpr 12795 . . . . . . . . . . . . . . . 16  |-  ( ( q  e.  Prime  /\  p  e.  Prime  /\  ( p  pCnt  A )  e.  NN0 )  ->  ( q  ||  ( p ^ (
p  pCnt  A )
)  ->  q  =  p ) )
3935, 36, 37, 38syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  ||  ( p ^ (
p  pCnt  A )
)  ->  q  =  p ) )
4039con3and 428 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  -.  q  ||  ( p ^
( p  pCnt  A
) ) )
41 prmnn 12761 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  Prime  ->  p  e.  NN )
4241adantl 452 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  p  e.  Prime )  ->  p  e.  NN )
4342, 19nnexpcld 11266 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN )
4443ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
p ^ ( p 
pCnt  A ) )  e.  NN )
45 pceq0 12923 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Prime  /\  (
p ^ ( p 
pCnt  A ) )  e.  NN )  ->  (
( q  pCnt  (
p ^ ( p 
pCnt  A ) ) )  =  0  <->  -.  q  ||  ( p ^ (
p  pCnt  A )
) ) )
4630, 44, 45syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
( q  pCnt  (
p ^ ( p 
pCnt  A ) ) )  =  0  <->  -.  q  ||  ( p ^ (
p  pCnt  A )
) ) )
4740, 46mpbird 223 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
q  pCnt  ( p ^ ( p  pCnt  A ) ) )  =  0 )
4834, 47eqtr4d 2318 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  p  e. 
Prime )  /\  (
q  e.  Prime  /\  (
q  ||  A  <->  q  =  p ) ) )  /\  -.  q  =  p )  ->  (
q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) )
4926, 48pm2.61dan 766 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  ( q  e.  Prime  /\  ( q  ||  A  <->  q  =  p ) ) )  ->  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p  pCnt  A ) ) ) )
5049expr 598 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  ( ( q 
||  A  <->  q  =  p )  ->  (
q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) ) )
5150ralimdva 2621 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  ->  A. q  e.  Prime  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) ) )
5251imp 418 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A. q  e.  Prime  ( q  pCnt  A )  =  ( q  pCnt  ( p ^ ( p 
pCnt  A ) ) ) )
53 nnnn0 9972 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  NN0 )
5453ad2antrr 706 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  e.  NN0 )
5543adantr 451 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN )
5655nnnn0d 10018 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( p ^ (
p  pCnt  A )
)  e.  NN0 )
57 pc11 12932 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  ( p ^ (
p  pCnt  A )
)  e.  NN0 )  ->  ( A  =  ( p ^ ( p 
pCnt  A ) )  <->  A. q  e.  Prime  ( q  pCnt  A )  =  ( q 
pCnt  ( p ^
( p  pCnt  A
) ) ) ) )
5854, 56, 57syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  -> 
( A  =  ( p ^ ( p 
pCnt  A ) )  <->  A. q  e.  Prime  ( q  pCnt  A )  =  ( q 
pCnt  ( p ^
( p  pCnt  A
) ) ) ) )
5952, 58mpbird 223 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  A  =  ( p ^ ( p  pCnt  A ) ) )
60 oveq2 5866 . . . . . . . . 9  |-  ( k  =  ( p  pCnt  A )  ->  ( p ^ k )  =  ( p ^ (
p  pCnt  A )
) )
6160eqeq2d 2294 . . . . . . . 8  |-  ( k  =  ( p  pCnt  A )  ->  ( A  =  ( p ^
k )  <->  A  =  ( p ^ (
p  pCnt  A )
) ) )
6261rspcev 2884 . . . . . . 7  |-  ( ( ( p  pCnt  A
)  e.  NN  /\  A  =  ( p ^ ( p  pCnt  A ) ) )  ->  E. k  e.  NN  A  =  ( p ^ k ) )
6314, 59, 62syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) )  ->  E. k  e.  NN  A  =  ( p ^ k ) )
6463ex 423 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  ->  E. k  e.  NN  A  =  ( p ^ k ) ) )
65 prmdvdsexpb 12794 . . . . . . . . . . 11  |-  ( ( q  e.  Prime  /\  p  e.  Prime  /\  k  e.  NN )  ->  ( q 
||  ( p ^
k )  <->  q  =  p ) )
66653coml 1158 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  k  e.  NN  /\  q  e. 
Prime )  ->  ( q 
||  ( p ^
k )  <->  q  =  p ) )
67663expa 1151 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  q  e.  Prime )  ->  ( q  ||  ( p ^ k
)  <->  q  =  p ) )
6867ralrimiva 2626 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  k  e.  NN )  ->  A. q  e.  Prime  ( q  ||  ( p ^ k
)  <->  q  =  p ) )
6968adantll 694 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  A. q  e.  Prime  ( q  ||  ( p ^ k )  <->  q  =  p ) )
70 breq2 4027 . . . . . . . . 9  |-  ( A  =  ( p ^
k )  ->  (
q  ||  A  <->  q  ||  ( p ^ k
) ) )
7170bibi1d 310 . . . . . . . 8  |-  ( A  =  ( p ^
k )  ->  (
( q  ||  A  <->  q  =  p )  <->  ( q  ||  ( p ^ k
)  <->  q  =  p ) ) )
7271ralbidv 2563 . . . . . . 7  |-  ( A  =  ( p ^
k )  ->  ( A. q  e.  Prime  ( q  ||  A  <->  q  =  p )  <->  A. q  e.  Prime  ( q  ||  ( p ^ k
)  <->  q  =  p ) ) )
7369, 72syl5ibrcom 213 . . . . . 6  |-  ( ( ( A  e.  NN  /\  p  e.  Prime )  /\  k  e.  NN )  ->  ( A  =  ( p ^ k
)  ->  A. q  e.  Prime  ( q  ||  A 
<->  q  =  p ) ) )
7473rexlimdva 2667 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( E. k  e.  NN  A  =  ( p ^ k )  ->  A. q  e.  Prime  ( q  ||  A  <->  q  =  p ) ) )
7564, 74impbid 183 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  Prime )  -> 
( A. q  e. 
Prime  ( q  ||  A  <->  q  =  p )  <->  E. k  e.  NN  A  =  ( p ^ k ) ) )
7675rexbidva 2560 . . 3  |-  ( A  e.  NN  ->  ( E. p  e.  Prime  A. q  e.  Prime  (
q  ||  A  <->  q  =  p )  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^ k ) ) )
772, 76syl5bb 248 . 2  |-  ( A  e.  NN  ->  ( E! q  e.  Prime  q 
||  A  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^ k ) ) )
781, 77bitrd 244 1  |-  ( A  e.  NN  ->  (
(Λ `  A )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  A  =  ( p ^
k ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0cc0 8737   NNcn 9746   NN0cn0 9965   ZZcz 10024   ^cexp 11104    || cdivides 12531   Primecprime 12758    pCnt cpc 12889  Λcvma 20329
This theorem is referenced by:  vmacl  20356  efvmacl  20358  vma1  20404  vmalelog  20444  fsumvma  20452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-vma 20335
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