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Theorem logtayllem 20006
Description: Lemma for logtayl 20007. (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 10262 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 9981 . . 3  |-  1  e.  NN0
32a1i 10 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  NN0 )
4 oveq2 5866 . . . . 5  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
5 eqid 2283 . . . . 5  |-  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )
6 ovex 5883 . . . . 5  |-  ( ( abs `  A ) ^ k )  e. 
_V
74, 5, 6fvmpt 5602 . . . 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 11763 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
109adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  RR )
11 reexpcl 11120 . . . 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 2357 . 2  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  e.  RR )
14 eqeq1 2289 . . . . . . 7  |-  ( n  =  k  ->  (
n  =  0  <->  k  =  0 ) )
15 oveq2 5866 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
1614, 15ifbieq2d 3585 . . . . . 6  |-  ( n  =  k  ->  if ( n  =  0 ,  0 ,  ( 1  /  n ) )  =  if ( k  =  0 ,  0 ,  ( 1  /  k ) ) )
17 oveq2 5866 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
1816, 17oveq12d 5876 . . . . 5  |-  ( n  =  k  ->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) )  =  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) ) )
19 eqid 2283 . . . . 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 5883 . . . . 5  |-  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^ k ) )  e.  _V
2118, 19, 20fvmpt 5602 . . . 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 8831 . . . . . 6  |-  0  e.  CC
2423a1i 10 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  ( abs `  A )  <  1
)  /\  k  e.  NN0 )  /\  k  =  0 )  ->  0  e.  CC )
25 nn0cn 9975 . . . . . . 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 2448 . . . . . . 7  |-  ( k  =/=  0  <->  -.  k  =  0 )
2827biimpri 197 . . . . . 6  |-  ( -.  k  =  0  -> 
k  =/=  0 )
29 reccl 9431 . . . . . 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 3592 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  if ( k  =  0 ,  0 ,  ( 1  / 
k ) )  e.  CC )
32 expcl 11121 . . . . 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 8855 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) )  e.  CC )
3522, 34eqeltrd 2357 . 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 8861 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  CC )
37 absidm 11807 . . . . . 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 4043 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  ( abs `  A ) )  <  1 )
4136, 40, 8geolim 12326 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  ~~>  ( 1  /  ( 1  -  ( abs `  A
) ) ) )
42 seqex 11048 . . . 4  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) )  e.  _V
43 ovex 5883 . . . 4  |-  ( 1  /  ( 1  -  ( abs `  A
) ) )  e. 
_V
4442, 43breldm 4883 . . 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 8837 . . 3  |-  1  e.  RR
4746a1i 10 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  RR )
48 elnnuz 10264 . . 3  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
49 nnrecre 9782 . . . . . . . . 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 8861 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  CC )
52 nnnn0 9972 . . . . . . . 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 11936 . . . . . 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 10363 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR+ )
5655adantl 452 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR+ )
5756rpreccld 10400 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  RR+ )
5857rpge0d 10394 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
1  /  k ) )
5950, 58absidd 11905 . . . . . . 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 11789 . . . . . . . 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 5876 . . . . . 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 2315 . . . . 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 11926 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  ( abs `  ( A ^
k ) ) )
6867, 62breqtrd 4047 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
( abs `  A
) ^ k ) )
69 nnge1 9772 . . . . . . . . 9  |-  ( k  e.  NN  ->  1  <_  k )
7069adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  1  <_  k
)
71 0lt1 9296 . . . . . . . . . 10  |-  0  <  1
7271a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  1
)
73 nnre 9753 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
7473adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR )
75 nngt0 9775 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
7675adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  k
)
77 lerec 9638 . . . . . . . . 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 8795 . . . . . . . 8  |-  1  e.  CC
8180div1i 9488 . . . . . . 7  |-  ( 1  /  1 )  =  1
8279, 81syl6breq 4062 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  <_  1
)
8350, 65, 66, 68, 82lemul1ad 9696 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( 1  /  k )  x.  ( ( abs `  A
) ^ k ) )  <_  ( 1  x.  ( ( abs `  A ) ^ k
) ) )
8464, 83eqbrtrd 4043 . . . 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 9778 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  =/=  0 )
8786adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  =/=  0
)
8887neneqd 2462 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  -.  k  = 
0 )
89 iffalse 3572 . . . . . . . 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 5873 . . . . . 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 2315 . . . . 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 5529 . . . 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 5874 . . . 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 4053 . . 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 12276 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 1623    e. wcel 1684    =/= wne 2446   ifcif 3565   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZ>=cuz 10230   RR+crp 10354    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  logtayl  20007
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-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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  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
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