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Theorem cxp2limlem 20286
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 8854 . . 3  |-  0  e.  RR
21a1i 10 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
3 2rp 10375 . . . . 5  |-  2  e.  RR+
4 rplogcl 19974 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
5 2z 10070 . . . . . 6  |-  2  e.  ZZ
6 rpexpcl 11138 . . . . . 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 10392 . . . . 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 10408 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  CC )
11 divrcnv 12327 . . 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 10406 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR )
14 rerpdivcl 10397 . . 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 8853 . . . . . . . 8  |-  1  e.  RR
1918a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
20 0lt1 9312 . . . . . . . 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 8993 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
2417, 23elrpd 10404 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
25 rpre 10376 . . . . 5  |-  ( n  e.  RR+  ->  n  e.  RR )
26 rpcxpcl 20039 . . . . 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 10423 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  e.  RR+ )
2928rpred 10406 . 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 10422 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR+ )
3231rpred 10406 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR )
3332resqcld 11287 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  e.  RR )
3433rehalfcld 9974 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  e.  RR )
35 1rp 10374 . . . . . . . . . . 11  |-  1  e.  RR+
36 rpaddcl 10390 . . . . . . . . . . 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 10406 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR )
3938, 34readdcld 8878 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )  e.  RR )
4032reefcld 12385 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( exp `  (
n  x.  ( log `  A ) ) )  e.  RR )
4134, 37ltaddrp2d 10436 . . . . . . . 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 12410 . . . . . . . . 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 8993 . . . . . . 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 8877 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  CC )
4746sqcld 11259 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  CC )
48 2cn 9832 . . . . . . . . . 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 10408 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  CC )
52 2ne0 9845 . . . . . . . . . 10  |-  2  =/=  0
5352a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  =/=  0
)
5450rpne0d 10411 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  =/=  0 )
5547, 49, 51, 53, 54divdiv2d 9584 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n ^
2 )  x.  (
( log `  A
) ^ 2 ) )  /  2 ) )
564rpcnd 10408 . . . . . . . . . . 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 11273 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  =  ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) ) )
5958oveq1d 5889 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  =  ( ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) )  /  2 ) )
6055, 59eqtr4d 2331 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )
6117recnd 8877 . . . . . . . . 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 10411 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  =/=  0
)
6562, 64, 46cxpefd 20075 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^ c  n )  =  ( exp `  ( n  x.  ( log `  A
) ) ) )
6644, 60, 653brtr4d 4069 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  <  ( A  ^ c  n ) )
67 rpexpcl 11138 . . . . . . . . 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 10423 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  e.  RR+ )
7170, 27, 16ltdiv2d 10429 . . . . . 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 10411 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =/=  0
)
7569rpne0d 10411 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  =/=  0 )
7646, 47, 73, 74, 75divdiv2d 9584 . . . . . 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 11258 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =  ( n  x.  n ) )
7877oveq2d 5890 . . . . . 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 10385 . . . . . . . 8  |-  ( n  e.  RR+  ->  n  =/=  0 )
8079adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  =/=  0
)
8173, 46, 46, 80, 80divcan5d 9578 . . . . . 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 2332 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( ( n ^
2 )  /  (
2  /  ( ( log `  A ) ^ 2 ) ) ) )  =  ( ( 2  /  (
( log `  A
) ^ 2 ) )  /  n ) )
8372, 82breqtrd 4063 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^ c  n ) )  < 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8429, 15, 83ltled 8983 . . 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 10410 . . 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 12140 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 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439   2c2 9811   ZZcz 10040   RR+crp 10370   ^cexp 11120    ~~> r crli 11975   expce 12359   logclog 19928    ^ c ccxp 19929
This theorem is referenced by:  cxp2lim  20287  cxploglim  20288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931
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