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Theorem cxp2limlem 20266
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 8835 . . 3  |-  0  e.  RR
21a1i 12 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
3 2rp 10356 . . . . 5  |-  2  e.  RR+
4 rplogcl 19954 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
5 2z 10051 . . . . . 6  |-  2  e.  ZZ
6 rpexpcl 11118 . . . . . 6  |-  ( ( ( log `  A
)  e.  RR+  /\  2  e.  ZZ )  ->  (
( log `  A
) ^ 2 )  e.  RR+ )
74, 5, 6sylancl 645 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( ( log `  A
) ^ 2 )  e.  RR+ )
8 rpdivcl 10373 . . . . 5  |-  ( ( 2  e.  RR+  /\  (
( log `  A
) ^ 2 )  e.  RR+ )  ->  (
2  /  ( ( log `  A ) ^ 2 ) )  e.  RR+ )
93, 7, 8sylancr 646 . . . 4  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR+ )
109rpcnd 10389 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  CC )
11 divrcnv 12307 . . 3  |-  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  e.  CC  ->  (
n  e.  RR+  |->  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )  ~~> r  0 )
1210, 11syl 17 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( n  e.  RR+  |->  ( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )  ~~> r  0 )
139rpred 10387 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR )
14 rerpdivcl 10378 . . 3  |-  ( ( ( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR  /\  n  e.  RR+ )  -> 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n )  e.  RR )
1513, 14sylan 459 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n
)  e.  RR )
16 simpr 449 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  RR+ )
17 simpl 445 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
18 1re 8834 . . . . . . . 8  |-  1  e.  RR
1918a1i 12 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
20 0lt1 9293 . . . . . . . 8  |-  0  <  1
2120a1i 12 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
22 simpr 449 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
232, 19, 17, 21, 22lttrd 8974 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
2417, 23elrpd 10385 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
25 rpre 10357 . . . . 5  |-  ( n  e.  RR+  ->  n  e.  RR )
26 rpcxpcl 20019 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR )  ->  ( A  ^ c  n )  e.  RR+ )
2724, 25, 26syl2an 465 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^ c  n )  e.  RR+ )
2816, 27rpdivcld 10404 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  e.  RR+ )
2928rpred 10387 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  e.  RR )
304adantr 453 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( log `  A
)  e.  RR+ )
3116, 30rpmulcld 10403 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR+ )
3231rpred 10387 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR )
3332resqcld 11267 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  e.  RR )
3433rehalfcld 9955 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  e.  RR )
35 1rp 10355 . . . . . . . . . . 11  |-  1  e.  RR+
36 rpaddcl 10371 . . . . . . . . . . 11  |-  ( ( 1  e.  RR+  /\  (
n  x.  ( log `  A ) )  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A
) ) )  e.  RR+ )
3735, 31, 36sylancr 646 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR+ )
3837rpred 10387 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR )
3938, 34readdcld 8859 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )  e.  RR )
4032reefcld 12365 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( exp `  (
n  x.  ( log `  A ) ) )  e.  RR )
4134, 37ltaddrp2d 10417 . . . . . . . 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 12390 . . . . . . . . 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 17 . . . . . . . 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 8974 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  <  ( exp `  ( n  x.  ( log `  A ) ) ) )
4525adantl 454 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  RR )
4645recnd 8858 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  CC )
4746sqcld 11239 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  CC )
48 2cn 9813 . . . . . . . . . 10  |-  2  e.  CC
4948a1i 12 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  e.  CC )
507adantr 453 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  RR+ )
5150rpcnd 10389 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  CC )
52 2ne0 9826 . . . . . . . . . 10  |-  2  =/=  0
5352a1i 12 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  =/=  0
)
5450rpne0d 10392 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  =/=  0 )
5547, 49, 51, 53, 54divdiv2d 9565 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n ^
2 )  x.  (
( log `  A
) ^ 2 ) )  /  2 ) )
564rpcnd 10389 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  CC )
5756adantr 453 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( log `  A
)  e.  CC )
5846, 57sqmuld 11253 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  =  ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) ) )
5958oveq1d 5836 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  =  ( ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) )  /  2 ) )
6055, 59eqtr4d 2321 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )
6117recnd 8858 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  CC )
6261adantr 453 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  e.  CC )
6324adantr 453 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  e.  RR+ )
6463rpne0d 10392 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  =/=  0
)
6562, 64, 46cxpefd 20055 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^ c  n )  =  ( exp `  ( n  x.  ( log `  A
) ) ) )
6644, 60, 653brtr4d 4056 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  <  ( A  ^ c  n ) )
67 rpexpcl 11118 . . . . . . . . 9  |-  ( ( n  e.  RR+  /\  2  e.  ZZ )  ->  (
n ^ 2 )  e.  RR+ )
6816, 5, 67sylancl 645 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  RR+ )
699adantr 453 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  e.  RR+ )
7068, 69rpdivcld 10404 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  e.  RR+ )
7170, 27, 16ltdiv2d 10410 . . . . . 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 203 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  < 
( n  /  (
( n ^ 2 )  /  ( 2  /  ( ( log `  A ) ^ 2 ) ) ) ) )
7310adantr 453 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  e.  CC )
7468rpne0d 10392 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =/=  0
)
7569rpne0d 10392 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  =/=  0 )
7646, 47, 73, 74, 75divdiv2d 9565 . . . . . 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 11238 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =  ( n  x.  n ) )
7877oveq2d 5837 . . . . . 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 10366 . . . . . . . 8  |-  ( n  e.  RR+  ->  n  =/=  0 )
8079adantl 454 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  =/=  0
)
8173, 46, 46, 80, 80divcan5d 9559 . . . . . 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 2322 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( ( n ^
2 )  /  (
2  /  ( ( log `  A ) ^ 2 ) ) ) )  =  ( ( 2  /  (
( log `  A
) ^ 2 ) )  /  n ) )
8372, 82breqtrd 4050 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  < 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8429, 15, 83ltled 8964 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  <_ 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8584adantrr 699 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( n  e.  RR+  /\  0  <_  n
) )  ->  (
n  /  ( A  ^ c  n ) )  <_  ( (
2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )
8628rpge0d 10391 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  0  <_  (
n  /  ( A  ^ c  n ) ) )
8786adantrr 699 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( n  e.  RR+  /\  0  <_  n
) )  ->  0  <_  ( n  /  ( A  ^ c  n ) ) )
882, 2, 12, 15, 29, 85, 87rlimsqz2 12120 1  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( n  e.  RR+  |->  ( n  /  ( A  ^ c  n ) ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1687    =/= wne 2449   class class class wbr 4026    e. cmpt 4080   ` cfv 5223  (class class class)co 5821   CCcc 8732   RRcr 8733   0cc0 8734   1c1 8735    + caddc 8737    x. cmul 8739    < clt 8864    <_ cle 8865    / cdiv 9420   2c2 9792   ZZcz 10021   RR+crp 10351   ^cexp 11100    ~~> r crli 11955   expce 12339   logclog 19908    ^ c ccxp 19909
This theorem is referenced by:  cxp2lim  20267  cxploglim  20268
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812  ax-addf 8813  ax-mulf 8814
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-of 6041  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-er 6657  df-map 6771  df-pm 6772  df-ixp 6815  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-fi 7162  df-sup 7191  df-oi 7222  df-card 7569  df-cda 7791  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-4 9803  df-5 9804  df-6 9805  df-7 9806  df-8 9807  df-9 9808  df-10 9809  df-n0 9963  df-z 10022  df-dec 10122  df-uz 10228  df-q 10314  df-rp 10352  df-xneg 10449  df-xadd 10450  df-xmul 10451  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-sum 12155  df-ef 12345  df-sin 12347  df-cos 12348  df-pi 12350  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-limc 19212  df-dv 19213  df-log 19910  df-cxp 19911
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