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Theorem logtayllem 20503
Description: Lemma for logtayl 20504. (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 10476 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 10193 . . 3  |-  1  e.  NN0
32a1i 11 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  NN0 )
4 oveq2 6048 . . . . 5  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
5 eqid 2404 . . . . 5  |-  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )
6 ovex 6065 . . . . 5  |-  ( ( abs `  A ) ^ k )  e. 
_V
74, 5, 6fvmpt 5765 . . . 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 12038 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
109adantr 452 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  RR )
11 reexpcl 11353 . . . 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 2478 . 2  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  e.  RR )
14 eqeq1 2410 . . . . . . 7  |-  ( n  =  k  ->  (
n  =  0  <->  k  =  0 ) )
15 oveq2 6048 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
1614, 15ifbieq2d 3719 . . . . . 6  |-  ( n  =  k  ->  if ( n  =  0 ,  0 ,  ( 1  /  n ) )  =  if ( k  =  0 ,  0 ,  ( 1  /  k ) ) )
17 oveq2 6048 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
1816, 17oveq12d 6058 . . . . 5  |-  ( n  =  k  ->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) )  =  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) ) )
19 eqid 2404 . . . . 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 6065 . . . . 5  |-  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^ k ) )  e.  _V
2118, 19, 20fvmpt 5765 . . . 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 9040 . . . . . 6  |-  0  e.  CC
2423a1i 11 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  ( abs `  A )  <  1
)  /\  k  e.  NN0 )  /\  k  =  0 )  ->  0  e.  CC )
25 nn0cn 10187 . . . . . . 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 2569 . . . . . . 7  |-  ( k  =/=  0  <->  -.  k  =  0 )
2827biimpri 198 . . . . . 6  |-  ( -.  k  =  0  -> 
k  =/=  0 )
29 reccl 9641 . . . . . 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 3726 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  if ( k  =  0 ,  0 ,  ( 1  / 
k ) )  e.  CC )
32 expcl 11354 . . . . 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 9064 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) )  e.  CC )
3522, 34eqeltrd 2478 . 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 9070 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  CC )
37 absidm 12082 . . . . . 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 4192 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  ( abs `  A ) )  <  1 )
4136, 40, 8geolim 12602 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  ~~>  ( 1  /  ( 1  -  ( abs `  A
) ) ) )
42 seqex 11280 . . . 4  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) )  e.  _V
43 ovex 6065 . . . 4  |-  ( 1  /  ( 1  -  ( abs `  A
) ) )  e. 
_V
4442, 43breldm 5033 . . 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 9046 . . 3  |-  1  e.  RR
4746a1i 11 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  RR )
48 elnnuz 10478 . . 3  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
49 nnrecre 9992 . . . . . . . . 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 9070 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  CC )
52 nnnn0 10184 . . . . . . . 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 12211 . . . . . 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 10577 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR+ )
5655adantl 453 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR+ )
5756rpreccld 10614 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  RR+ )
5857rpge0d 10608 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
1  /  k ) )
5950, 58absidd 12180 . . . . . . 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 12064 . . . . . . . 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 6058 . . . . . 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 2436 . . . . 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 12201 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  ( abs `  ( A ^
k ) ) )
6867, 62breqtrd 4196 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
( abs `  A
) ^ k ) )
69 nnge1 9982 . . . . . . . . 9  |-  ( k  e.  NN  ->  1  <_  k )
7069adantl 453 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  1  <_  k
)
71 0lt1 9506 . . . . . . . . . 10  |-  0  <  1
7271a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  1
)
73 nnre 9963 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
7473adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR )
75 nngt0 9985 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
7675adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  k
)
77 lerec 9848 . . . . . . . . 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 9004 . . . . . . . 8  |-  1  e.  CC
8180div1i 9698 . . . . . . 7  |-  ( 1  /  1 )  =  1
8279, 81syl6breq 4211 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  <_  1
)
8350, 65, 66, 68, 82lemul1ad 9906 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( 1  /  k )  x.  ( ( abs `  A
) ^ k ) )  <_  ( 1  x.  ( ( abs `  A ) ^ k
) ) )
8464, 83eqbrtrd 4192 . . . 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 9988 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  =/=  0 )
8786adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  =/=  0
)
8887neneqd 2583 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  -.  k  = 
0 )
89 iffalse 3706 . . . . . . . 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 6055 . . . . . 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 2436 . . . . 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 5691 . . . 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 6056 . . . 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 4202 . . 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 12552 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 1721    =/= wne 2567   ifcif 3699   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZ>=cuz 10444   RR+crp 10568    seq cseq 11278   ^cexp 11337   abscabs 11994    ~~> cli 12233
This theorem is referenced by:  logtayl  20504
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-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  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  ax-addf 9025  ax-mulf 9026
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-se 4502  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-isom 5422  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-oadd 6687  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  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-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435
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