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Theorem logtayllem 20059
Description: Lemma for logtayl 20060. (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 10309 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 10028 . . 3  |-  1  e.  NN0
32a1i 10 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  NN0 )
4 oveq2 5908 . . . . 5  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
5 eqid 2316 . . . . 5  |-  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )
6 ovex 5925 . . . . 5  |-  ( ( abs `  A ) ^ k )  e. 
_V
74, 5, 6fvmpt 5640 . . . 4  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  =  ( ( abs `  A
) ^ k ) )
87adantl 452 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  =  ( ( abs `  A
) ^ k ) )
9 abscl 11810 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
109adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  RR )
11 reexpcl 11167 . . . 4  |-  ( ( ( abs `  A
)  e.  RR  /\  k  e.  NN0 )  -> 
( ( abs `  A
) ^ k )  e.  RR )
1210, 11sylan 457 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( abs `  A ) ^ k
)  e.  RR )
138, 12eqeltrd 2390 . 2  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  e.  RR )
14 eqeq1 2322 . . . . . . 7  |-  ( n  =  k  ->  (
n  =  0  <->  k  =  0 ) )
15 oveq2 5908 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
1614, 15ifbieq2d 3619 . . . . . 6  |-  ( n  =  k  ->  if ( n  =  0 ,  0 ,  ( 1  /  n ) )  =  if ( k  =  0 ,  0 ,  ( 1  /  k ) ) )
17 oveq2 5908 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
1816, 17oveq12d 5918 . . . . 5  |-  ( n  =  k  ->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) )  =  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) ) )
19 eqid 2316 . . . . 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 5925 . . . . 5  |-  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^ k ) )  e.  _V
2118, 19, 20fvmpt 5640 . . . 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 452 . . 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 8876 . . . . . 6  |-  0  e.  CC
2423a1i 10 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  ( abs `  A )  <  1
)  /\  k  e.  NN0 )  /\  k  =  0 )  ->  0  e.  CC )
25 nn0cn 10022 . . . . . . 7  |-  ( k  e.  NN0  ->  k  e.  CC )
2625adantl 452 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  k  e.  CC )
27 df-ne 2481 . . . . . . 7  |-  ( k  =/=  0  <->  -.  k  =  0 )
2827biimpri 197 . . . . . 6  |-  ( -.  k  =  0  -> 
k  =/=  0 )
29 reccl 9476 . . . . . 6  |-  ( ( k  e.  CC  /\  k  =/=  0 )  -> 
( 1  /  k
)  e.  CC )
3026, 28, 29syl2an 463 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  ( abs `  A )  <  1
)  /\  k  e.  NN0 )  /\  -.  k  =  0 )  -> 
( 1  /  k
)  e.  CC )
3124, 30ifclda 3626 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  if ( k  =  0 ,  0 ,  ( 1  / 
k ) )  e.  CC )
32 expcl 11168 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
3332adantlr 695 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( A ^
k )  e.  CC )
3431, 33mulcld 8900 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) )  e.  CC )
3522, 34eqeltrd 2390 . 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 8906 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  CC )
37 absidm 11854 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( abs `  A
) )  =  ( abs `  A ) )
3837adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  ( abs `  A ) )  =  ( abs `  A
) )
39 simpr 447 . . . . 5  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  <  1 )
4038, 39eqbrtrd 4080 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  ( abs `  A ) )  <  1 )
4136, 40, 8geolim 12373 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  ~~>  ( 1  /  ( 1  -  ( abs `  A
) ) ) )
42 seqex 11095 . . . 4  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) )  e.  _V
43 ovex 5925 . . . 4  |-  ( 1  /  ( 1  -  ( abs `  A
) ) )  e. 
_V
4442, 43breldm 4920 . . 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 15 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  e.  dom  ~~>  )
46 1re 8882 . . 3  |-  1  e.  RR
4746a1i 10 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  RR )
48 elnnuz 10311 . . 3  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
49 nnrecre 9827 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
5049adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  RR )
5150recnd 8906 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  CC )
52 nnnn0 10019 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
5352, 33sylan2 460 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( A ^
k )  e.  CC )
5451, 53absmuld 11983 . . . . . 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 10410 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR+ )
5655adantl 452 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR+ )
5756rpreccld 10447 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  RR+ )
5857rpge0d 10441 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
1  /  k ) )
5950, 58absidd 11952 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
1  /  k ) )  =  ( 1  /  k ) )
60 simpl 443 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  A  e.  CC )
61 absexp 11836 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )
6260, 52, 61syl2an 463 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )
6359, 62oveq12d 5918 . . . . . 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 2348 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
( 1  /  k
)  x.  ( A ^ k ) ) )  =  ( ( 1  /  k )  x.  ( ( abs `  A ) ^ k
) ) )
6546a1i 10 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  1  e.  RR )
6652, 12sylan2 460 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( abs `  A ) ^ k
)  e.  RR )
6753absge0d 11973 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  ( abs `  ( A ^
k ) ) )
6867, 62breqtrd 4084 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
( abs `  A
) ^ k ) )
69 nnge1 9817 . . . . . . . . 9  |-  ( k  e.  NN  ->  1  <_  k )
7069adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  1  <_  k
)
71 0lt1 9341 . . . . . . . . . 10  |-  0  <  1
7271a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  1
)
73 nnre 9798 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
7473adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR )
75 nngt0 9820 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
7675adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  k
)
77 lerec 9683 . . . . . . . . 9  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( k  e.  RR  /\  0  < 
k ) )  -> 
( 1  <_  k  <->  ( 1  /  k )  <_  ( 1  / 
1 ) ) )
7865, 72, 74, 76, 77syl22anc 1183 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  <_ 
k  <->  ( 1  / 
k )  <_  (
1  /  1 ) ) )
7970, 78mpbid 201 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  <_  (
1  /  1 ) )
80 ax-1cn 8840 . . . . . . . 8  |-  1  e.  CC
8180div1i 9533 . . . . . . 7  |-  ( 1  /  1 )  =  1
8279, 81syl6breq 4099 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  <_  1
)
8350, 65, 66, 68, 82lemul1ad 9741 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( 1  /  k )  x.  ( ( abs `  A
) ^ k ) )  <_  ( 1  x.  ( ( abs `  A ) ^ k
) ) )
8464, 83eqbrtrd 4080 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( abs `  (
( 1  /  k
)  x.  ( A ^ k ) ) )  <_  ( 1  x.  ( ( abs `  A ) ^ k
) ) )
8552, 22sylan2 460 . . . . . 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 9823 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  =/=  0 )
8786adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  =/=  0
)
8887neneqd 2495 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  -.  k  = 
0 )
89 iffalse 3606 . . . . . . . 8  |-  ( -.  k  =  0  ->  if ( k  =  0 ,  0 ,  ( 1  /  k ) )  =  ( 1  /  k ) )
9088, 89syl 15 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  if ( k  =  0 ,  0 ,  ( 1  / 
k ) )  =  ( 1  /  k
) )
9190oveq1d 5915 . . . . . 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 2348 . . . . 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 5567 . . . 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 460 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  =  ( ( abs `  A
) ^ k ) )
9594oveq2d 5916 . . . 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 4090 . . 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 462 . 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 12323 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 176    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479   ifcif 3599   class class class wbr 4060    e. cmpt 4114   dom cdm 4726   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    < clt 8912    <_ cle 8913    - cmin 9082    / cdiv 9468   NNcn 9791   NN0cn0 10012   ZZ>=cuz 10277   RR+crp 10401    seq cseq 11093   ^cexp 11151   abscabs 11766    ~~> cli 12005
This theorem is referenced by:  logtayl  20060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-pm 6818  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-ico 10709  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-sum 12206
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