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Theorem cxp2limlem 20270
Description: A linear factor grows slower than any exponential with base greater than  1. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
cxp2limlem  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( n  e.  RR+  |->  ( n  /  ( A  ^ c  n ) ) )  ~~> r  0 )
Distinct variable group:    A, n

Proof of Theorem cxp2limlem
StepHypRef Expression
1 0re 8838 . . 3  |-  0  e.  RR
21a1i 10 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
3 2rp 10359 . . . . 5  |-  2  e.  RR+
4 rplogcl 19958 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
5 2z 10054 . . . . . 6  |-  2  e.  ZZ
6 rpexpcl 11122 . . . . . 6  |-  ( ( ( log `  A
)  e.  RR+  /\  2  e.  ZZ )  ->  (
( log `  A
) ^ 2 )  e.  RR+ )
74, 5, 6sylancl 643 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( ( log `  A
) ^ 2 )  e.  RR+ )
8 rpdivcl 10376 . . . . 5  |-  ( ( 2  e.  RR+  /\  (
( log `  A
) ^ 2 )  e.  RR+ )  ->  (
2  /  ( ( log `  A ) ^ 2 ) )  e.  RR+ )
93, 7, 8sylancr 644 . . . 4  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR+ )
109rpcnd 10392 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  CC )
11 divrcnv 12311 . . 3  |-  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  e.  CC  ->  (
n  e.  RR+  |->  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )  ~~> r  0 )
1210, 11syl 15 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( n  e.  RR+  |->  ( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )  ~~> r  0 )
139rpred 10390 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR )
14 rerpdivcl 10381 . . 3  |-  ( ( ( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR  /\  n  e.  RR+ )  -> 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n )  e.  RR )
1513, 14sylan 457 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n
)  e.  RR )
16 simpr 447 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  RR+ )
17 simpl 443 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
18 1re 8837 . . . . . . . 8  |-  1  e.  RR
1918a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
20 0lt1 9296 . . . . . . . 8  |-  0  <  1
2120a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
22 simpr 447 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
232, 19, 17, 21, 22lttrd 8977 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
2417, 23elrpd 10388 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
25 rpre 10360 . . . . 5  |-  ( n  e.  RR+  ->  n  e.  RR )
26 rpcxpcl 20023 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR )  ->  ( A  ^ c  n )  e.  RR+ )
2724, 25, 26syl2an 463 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^ c  n )  e.  RR+ )
2816, 27rpdivcld 10407 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  e.  RR+ )
2928rpred 10390 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  e.  RR )
304adantr 451 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( log `  A
)  e.  RR+ )
3116, 30rpmulcld 10406 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR+ )
3231rpred 10390 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR )
3332resqcld 11271 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  e.  RR )
3433rehalfcld 9958 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  e.  RR )
35 1rp 10358 . . . . . . . . . . 11  |-  1  e.  RR+
36 rpaddcl 10374 . . . . . . . . . . 11  |-  ( ( 1  e.  RR+  /\  (
n  x.  ( log `  A ) )  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A
) ) )  e.  RR+ )
3735, 31, 36sylancr 644 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR+ )
3837rpred 10390 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR )
3938, 34readdcld 8862 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )  e.  RR )
4032reefcld 12369 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( exp `  (
n  x.  ( log `  A ) ) )  e.  RR )
4134, 37ltaddrp2d 10420 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  <  ( (
1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) ) )
42 efgt1p2 12394 . . . . . . . . 9  |-  ( ( n  x.  ( log `  A ) )  e.  RR+  ->  ( ( 1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )  <  ( exp `  ( n  x.  ( log `  A ) ) ) )
4331, 42syl 15 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )  <  ( exp `  ( n  x.  ( log `  A ) ) ) )
4434, 39, 40, 41, 43lttrd 8977 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  <  ( exp `  ( n  x.  ( log `  A ) ) ) )
4525adantl 452 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  RR )
4645recnd 8861 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  CC )
4746sqcld 11243 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  CC )
48 2cn 9816 . . . . . . . . . 10  |-  2  e.  CC
4948a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  e.  CC )
507adantr 451 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  RR+ )
5150rpcnd 10392 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  CC )
52 2ne0 9829 . . . . . . . . . 10  |-  2  =/=  0
5352a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  =/=  0
)
5450rpne0d 10395 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  =/=  0 )
5547, 49, 51, 53, 54divdiv2d 9568 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n ^
2 )  x.  (
( log `  A
) ^ 2 ) )  /  2 ) )
564rpcnd 10392 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  CC )
5756adantr 451 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( log `  A
)  e.  CC )
5846, 57sqmuld 11257 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  =  ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) ) )
5958oveq1d 5873 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  =  ( ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) )  /  2 ) )
6055, 59eqtr4d 2318 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )
6117recnd 8861 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  CC )
6261adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  e.  CC )
6324adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  e.  RR+ )
6463rpne0d 10395 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  =/=  0
)
6562, 64, 46cxpefd 20059 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^ c  n )  =  ( exp `  ( n  x.  ( log `  A
) ) ) )
6644, 60, 653brtr4d 4053 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  <  ( A  ^ c  n ) )
67 rpexpcl 11122 . . . . . . . . 9  |-  ( ( n  e.  RR+  /\  2  e.  ZZ )  ->  (
n ^ 2 )  e.  RR+ )
6816, 5, 67sylancl 643 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  RR+ )
699adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  e.  RR+ )
7068, 69rpdivcld 10407 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  e.  RR+ )
7170, 27, 16ltdiv2d 10413 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n ^ 2 )  /  ( 2  / 
( ( log `  A
) ^ 2 ) ) )  <  ( A  ^ c  n )  <-> 
( n  /  ( A  ^ c  n ) )  <  ( n  /  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) ) ) ) )
7266, 71mpbid 201 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  < 
( n  /  (
( n ^ 2 )  /  ( 2  /  ( ( log `  A ) ^ 2 ) ) ) ) )
7310adantr 451 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  e.  CC )
7468rpne0d 10395 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =/=  0
)
7569rpne0d 10395 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  =/=  0 )
7646, 47, 73, 74, 75divdiv2d 9568 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( ( n ^
2 )  /  (
2  /  ( ( log `  A ) ^ 2 ) ) ) )  =  ( ( n  x.  (
2  /  ( ( log `  A ) ^ 2 ) ) )  /  ( n ^ 2 ) ) )
7746sqvald 11242 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =  ( n  x.  n ) )
7877oveq2d 5874 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( 2  / 
( ( log `  A
) ^ 2 ) ) )  /  (
n ^ 2 ) )  =  ( ( n  x.  ( 2  /  ( ( log `  A ) ^ 2 ) ) )  / 
( n  x.  n
) ) )
79 rpne0 10369 . . . . . . . 8  |-  ( n  e.  RR+  ->  n  =/=  0 )
8079adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  =/=  0
)
8173, 46, 46, 80, 80divcan5d 9562 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( 2  / 
( ( log `  A
) ^ 2 ) ) )  /  (
n  x.  n ) )  =  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )
8276, 78, 813eqtrd 2319 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( ( n ^
2 )  /  (
2  /  ( ( log `  A ) ^ 2 ) ) ) )  =  ( ( 2  /  (
( log `  A
) ^ 2 ) )  /  n ) )
8372, 82breqtrd 4047 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  < 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8429, 15, 83ltled 8967 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  <_ 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8584adantrr 697 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( n  e.  RR+  /\  0  <_  n
) )  ->  (
n  /  ( A  ^ c  n ) )  <_  ( (
2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )
8628rpge0d 10394 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  0  <_  (
n  /  ( A  ^ c  n ) ) )
8786adantrr 697 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( n  e.  RR+  /\  0  <_  n
) )  ->  0  <_  ( n  /  ( A  ^ c  n ) ) )
882, 2, 12, 15, 29, 85, 87rlimsqz2 12124 1  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( n  e.  RR+  |->  ( n  /  ( A  ^ c  n ) ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   2c2 9795   ZZcz 10024   RR+crp 10354   ^cexp 11104    ~~> r crli 11959   expce 12343   logclog 19912    ^ c ccxp 19913
This theorem is referenced by:  cxp2lim  20271  cxploglim  20272
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-iin 3908  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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  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-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  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  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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