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Theorem logtayllem 20417
Description: Lemma for logtayl 20418. (Contributed by Mario Carneiro, 1-Apr-2015.)
Assertion
Ref Expression
logtayllem  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n
) )  x.  ( A ^ n ) ) ) )  e.  dom  ~~>  )
Distinct variable group:    A, n

Proof of Theorem logtayllem
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nn0uz 10452 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 10169 . . 3  |-  1  e.  NN0
32a1i 11 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  NN0 )
4 oveq2 6028 . . . . 5  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
5 eqid 2387 . . . . 5  |-  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )
6 ovex 6045 . . . . 5  |-  ( ( abs `  A ) ^ k )  e. 
_V
74, 5, 6fvmpt 5745 . . . 4  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  =  ( ( abs `  A
) ^ k ) )
87adantl 453 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  =  ( ( abs `  A
) ^ k ) )
9 abscl 12010 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
109adantr 452 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  RR )
11 reexpcl 11325 . . . 4  |-  ( ( ( abs `  A
)  e.  RR  /\  k  e.  NN0 )  -> 
( ( abs `  A
) ^ k )  e.  RR )
1210, 11sylan 458 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( abs `  A ) ^ k
)  e.  RR )
138, 12eqeltrd 2461 . 2  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  e.  RR )
14 eqeq1 2393 . . . . . . 7  |-  ( n  =  k  ->  (
n  =  0  <->  k  =  0 ) )
15 oveq2 6028 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
1614, 15ifbieq2d 3702 . . . . . 6  |-  ( n  =  k  ->  if ( n  =  0 ,  0 ,  ( 1  /  n ) )  =  if ( k  =  0 ,  0 ,  ( 1  /  k ) ) )
17 oveq2 6028 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
1816, 17oveq12d 6038 . . . . 5  |-  ( n  =  k  ->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) )  =  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) ) )
19 eqid 2387 . . . . 5  |-  ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) ) )  =  ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) ) )
20 ovex 6045 . . . . 5  |-  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^ k ) )  e.  _V
2118, 19, 20fvmpt 5745 . . . 4  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) ) ) `  k )  =  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) ) )
2221adantl 453 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) ) ) `  k )  =  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) ) )
23 0cn 9017 . . . . . 6  |-  0  e.  CC
2423a1i 11 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  ( abs `  A )  <  1
)  /\  k  e.  NN0 )  /\  k  =  0 )  ->  0  e.  CC )
25 nn0cn 10163 . . . . . . 7  |-  ( k  e.  NN0  ->  k  e.  CC )
2625adantl 453 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  k  e.  CC )
27 df-ne 2552 . . . . . . 7  |-  ( k  =/=  0  <->  -.  k  =  0 )
2827biimpri 198 . . . . . 6  |-  ( -.  k  =  0  -> 
k  =/=  0 )
29 reccl 9617 . . . . . 6  |-  ( ( k  e.  CC  /\  k  =/=  0 )  -> 
( 1  /  k
)  e.  CC )
3026, 28, 29syl2an 464 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  ( abs `  A )  <  1
)  /\  k  e.  NN0 )  /\  -.  k  =  0 )  -> 
( 1  /  k
)  e.  CC )
3124, 30ifclda 3709 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  if ( k  =  0 ,  0 ,  ( 1  / 
k ) )  e.  CC )
32 expcl 11326 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
3332adantlr 696 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( A ^
k )  e.  CC )
3431, 33mulcld 9041 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) )  e.  CC )
3522, 34eqeltrd 2461 . 2  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) ) ) `  k )  e.  CC )
3610recnd 9047 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  CC )
37 absidm 12054 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( abs `  A
) )  =  ( abs `  A ) )
3837adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  ( abs `  A ) )  =  ( abs `  A
) )
39 simpr 448 . . . . 5  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  <  1 )
4038, 39eqbrtrd 4173 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  ( abs `  A ) )  <  1 )
4136, 40, 8geolim 12574 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  ~~>  ( 1  /  ( 1  -  ( abs `  A
) ) ) )
42 seqex 11252 . . . 4  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) )  e.  _V
43 ovex 6045 . . . 4  |-  ( 1  /  ( 1  -  ( abs `  A
) ) )  e. 
_V
4442, 43breldm 5014 . . 3  |-  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  ~~>  ( 1  /  ( 1  -  ( abs `  A
) ) )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  e.  dom  ~~>  )
4541, 44syl 16 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  e.  dom  ~~>  )
46 1re 9023 . . 3  |-  1  e.  RR
4746a1i 11 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  RR )
48 elnnuz 10454 . . 3  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
49 nnrecre 9968 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
5049adantl 453 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  RR )
5150recnd 9047 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  CC )
52 nnnn0 10160 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
5352, 33sylan2 461 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( A ^
k )  e.  CC )
5451, 53absmuld 12183 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
( 1  /  k
)  x.  ( A ^ k ) ) )  =  ( ( abs `  ( 1  /  k ) )  x.  ( abs `  ( A ^ k ) ) ) )
55 nnrp 10553 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR+ )
5655adantl 453 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR+ )
5756rpreccld 10590 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  RR+ )
5857rpge0d 10584 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
1  /  k ) )
5950, 58absidd 12152 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
1  /  k ) )  =  ( 1  /  k ) )
60 simpl 444 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  A  e.  CC )
61 absexp 12036 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )
6260, 52, 61syl2an 464 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )
6359, 62oveq12d 6038 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( abs `  ( 1  /  k
) )  x.  ( abs `  ( A ^
k ) ) )  =  ( ( 1  /  k )  x.  ( ( abs `  A
) ^ k ) ) )
6454, 63eqtrd 2419 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
( 1  /  k
)  x.  ( A ^ k ) ) )  =  ( ( 1  /  k )  x.  ( ( abs `  A ) ^ k
) ) )
6546a1i 11 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  1  e.  RR )
6652, 12sylan2 461 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( abs `  A ) ^ k
)  e.  RR )
6753absge0d 12173 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  ( abs `  ( A ^
k ) ) )
6867, 62breqtrd 4177 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
( abs `  A
) ^ k ) )
69 nnge1 9958 . . . . . . . . 9  |-  ( k  e.  NN  ->  1  <_  k )
7069adantl 453 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  1  <_  k
)
71 0lt1 9482 . . . . . . . . . 10  |-  0  <  1
7271a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  1
)
73 nnre 9939 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
7473adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR )
75 nngt0 9961 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
7675adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  k
)
77 lerec 9824 . . . . . . . . 9  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( k  e.  RR  /\  0  < 
k ) )  -> 
( 1  <_  k  <->  ( 1  /  k )  <_  ( 1  / 
1 ) ) )
7865, 72, 74, 76, 77syl22anc 1185 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  <_ 
k  <->  ( 1  / 
k )  <_  (
1  /  1 ) ) )
7970, 78mpbid 202 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  <_  (
1  /  1 ) )
80 ax-1cn 8981 . . . . . . . 8  |-  1  e.  CC
8180div1i 9674 . . . . . . 7  |-  ( 1  /  1 )  =  1
8279, 81syl6breq 4192 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  <_  1
)
8350, 65, 66, 68, 82lemul1ad 9882 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( 1  /  k )  x.  ( ( abs `  A
) ^ k ) )  <_  ( 1  x.  ( ( abs `  A ) ^ k
) ) )
8464, 83eqbrtrd 4173 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
( 1  /  k
)  x.  ( A ^ k ) ) )  <_  ( 1  x.  ( ( abs `  A ) ^ k
) ) )
8552, 22sylan2 461 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) ) ) `  k )  =  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) ) )
86 nnne0 9964 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  =/=  0 )
8786adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  =/=  0
)
8887neneqd 2566 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  -.  k  = 
0 )
89 iffalse 3689 . . . . . . . 8  |-  ( -.  k  =  0  ->  if ( k  =  0 ,  0 ,  ( 1  /  k ) )  =  ( 1  /  k ) )
9088, 89syl 16 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  if ( k  =  0 ,  0 ,  ( 1  / 
k ) )  =  ( 1  /  k
) )
9190oveq1d 6035 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) )  =  ( ( 1  / 
k )  x.  ( A ^ k ) ) )
9285, 91eqtrd 2419 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) ) ) `  k )  =  ( ( 1  /  k )  x.  ( A ^ k
) ) )
9392fveq2d 5672 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n
) )  x.  ( A ^ n ) ) ) `  k ) )  =  ( abs `  ( ( 1  / 
k )  x.  ( A ^ k ) ) ) )
9452, 8sylan2 461 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  =  ( ( abs `  A
) ^ k ) )
9594oveq2d 6036 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  x.  ( ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) ) `  k
) )  =  ( 1  x.  ( ( abs `  A ) ^ k ) ) )
9684, 93, 953brtr4d 4183 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n
) )  x.  ( A ^ n ) ) ) `  k ) )  <_  ( 1  x.  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k ) ) )
9748, 96sylan2br 463 . 2  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  (
ZZ>= `  1 ) )  ->  ( abs `  (
( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n
) )  x.  ( A ^ n ) ) ) `  k ) )  <_  ( 1  x.  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k ) ) )
981, 3, 13, 35, 45, 47, 97cvgcmpce 12524 1  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( if ( n  =  0 ,  0 ,  ( 1  /  n
) )  x.  ( A ^ n ) ) ) )  e.  dom  ~~>  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   ifcif 3682   class class class wbr 4153    e. cmpt 4207   dom cdm 4818   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   NNcn 9932   NN0cn0 10153   ZZ>=cuz 10420   RR+crp 10544    seq cseq 11250   ^cexp 11309   abscabs 11966    ~~> cli 12205
This theorem is referenced by:  logtayl  20418
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-ico 10854  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-sum 12407
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