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Theorem logtayllem 20008
Description: Lemma for logtayl 20009. (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 10264 . 2  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 9983 . . 3  |-  1  e.  NN0
32a1i 10 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  NN0 )
4 oveq2 5868 . . . . 5  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
5 eqid 2285 . . . . 5  |-  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )
6 ovex 5885 . . . . 5  |-  ( ( abs `  A ) ^ k )  e. 
_V
74, 5, 6fvmpt 5604 . . . 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 11765 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
109adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  RR )
11 reexpcl 11122 . . . 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 2359 . 2  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) `
 k )  e.  RR )
14 eqeq1 2291 . . . . . . 7  |-  ( n  =  k  ->  (
n  =  0  <->  k  =  0 ) )
15 oveq2 5868 . . . . . . 7  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
1614, 15ifbieq2d 3587 . . . . . 6  |-  ( n  =  k  ->  if ( n  =  0 ,  0 ,  ( 1  /  n ) )  =  if ( k  =  0 ,  0 ,  ( 1  /  k ) ) )
17 oveq2 5868 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
1816, 17oveq12d 5878 . . . . 5  |-  ( n  =  k  ->  ( if ( n  =  0 ,  0 ,  ( 1  /  n ) )  x.  ( A ^ n ) )  =  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) ) )
19 eqid 2285 . . . . 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 5885 . . . . 5  |-  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^ k ) )  e.  _V
2118, 19, 20fvmpt 5604 . . . 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 8833 . . . . . 6  |-  0  e.  CC
2423a1i 10 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  ( abs `  A )  <  1
)  /\  k  e.  NN0 )  /\  k  =  0 )  ->  0  e.  CC )
25 nn0cn 9977 . . . . . . 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 2450 . . . . . . 7  |-  ( k  =/=  0  <->  -.  k  =  0 )
2827biimpri 197 . . . . . 6  |-  ( -.  k  =  0  -> 
k  =/=  0 )
29 reccl 9433 . . . . . 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 3594 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  if ( k  =  0 ,  0 ,  ( 1  / 
k ) )  e.  CC )
32 expcl 11123 . . . . 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 8857 . . 3  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN0 )  ->  ( if ( k  =  0 ,  0 ,  ( 1  /  k ) )  x.  ( A ^
k ) )  e.  CC )
3522, 34eqeltrd 2359 . 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 8863 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  A
)  e.  CC )
37 absidm 11809 . . . . . 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 4045 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( abs `  ( abs `  A ) )  <  1 )
4136, 40, 8geolim 12328 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) ) )  ~~>  ( 1  /  ( 1  -  ( abs `  A
) ) ) )
42 seqex 11050 . . . 4  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) ) )  e.  _V
43 ovex 5885 . . . 4  |-  ( 1  /  ( 1  -  ( abs `  A
) ) )  e. 
_V
4442, 43breldm 4885 . . 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 8839 . . 3  |-  1  e.  RR
4746a1i 10 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
1  e.  RR )
48 elnnuz 10266 . . 3  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
49 nnrecre 9784 . . . . . . . . 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 8863 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  CC )
52 nnnn0 9974 . . . . . . . 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 11938 . . . . . 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 10365 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR+ )
5655adantl 452 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR+ )
5756rpreccld 10402 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  e.  RR+ )
5857rpge0d 10396 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
1  /  k ) )
5950, 58absidd 11907 . . . . . . 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 11791 . . . . . . . 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 5878 . . . . . 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 2317 . . . . 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 11928 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  ( abs `  ( A ^
k ) ) )
6867, 62breqtrd 4049 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <_  (
( abs `  A
) ^ k ) )
69 nnge1 9774 . . . . . . . . 9  |-  ( k  e.  NN  ->  1  <_  k )
7069adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  1  <_  k
)
71 0lt1 9298 . . . . . . . . . 10  |-  0  <  1
7271a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  1
)
73 nnre 9755 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
7473adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  e.  RR )
75 nngt0 9777 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
7675adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  0  <  k
)
77 lerec 9640 . . . . . . . . 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 8797 . . . . . . . 8  |-  1  e.  CC
8180div1i 9490 . . . . . . 7  |-  ( 1  /  1 )  =  1
8279, 81syl6breq 4064 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( 1  / 
k )  <_  1
)
8350, 65, 66, 68, 82lemul1ad 9698 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  ( ( 1  /  k )  x.  ( ( abs `  A
) ^ k ) )  <_  ( 1  x.  ( ( abs `  A ) ^ k
) ) )
8464, 83eqbrtrd 4045 . . . 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 9780 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  =/=  0 )
8786adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  k  =/=  0
)
8887neneqd 2464 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  k  e.  NN )  ->  -.  k  = 
0 )
89 iffalse 3574 . . . . . . . 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 5875 . . . . . 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 2317 . . . . 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 5531 . . . 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 5876 . . . 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 4055 . . 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 12278 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 1625    e. wcel 1686    =/= wne 2448   ifcif 3567   class class class wbr 4025    e. cmpt 4079   dom cdm 4691   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    < clt 8869    <_ cle 8870    - cmin 9039    / cdiv 9425   NNcn 9748   NN0cn0 9967   ZZ>=cuz 10232   RR+crp 10356    seq cseq 11048   ^cexp 11106   abscabs 11721    ~~> cli 11960
This theorem is referenced by:  logtayl  20009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-pm 6777  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-ico 10664  df-fz 10785  df-fzo 10873  df-fl 10927  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161
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