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Theorem cxp2limlem 23887
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 0red 9644 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
2 2rp 11307 . . . . 5  |-  2  e.  RR+
3 rplogcl 23539 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
4 2z 10969 . . . . . 6  |-  2  e.  ZZ
5 rpexpcl 12290 . . . . . 6  |-  ( ( ( log `  A
)  e.  RR+  /\  2  e.  ZZ )  ->  (
( log `  A
) ^ 2 )  e.  RR+ )
63, 4, 5sylancl 666 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( ( log `  A
) ^ 2 )  e.  RR+ )
7 rpdivcl 11325 . . . . 5  |-  ( ( 2  e.  RR+  /\  (
( log `  A
) ^ 2 )  e.  RR+ )  ->  (
2  /  ( ( log `  A ) ^ 2 ) )  e.  RR+ )
82, 6, 7sylancr 667 . . . 4  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR+ )
98rpcnd 11343 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  CC )
10 divrcnv 13897 . . 3  |-  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  e.  CC  ->  (
n  e.  RR+  |->  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )  ~~> r  0 )
119, 10syl 17 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( n  e.  RR+  |->  ( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )  ~~> r  0 )
128rpred 11341 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR )
13 rerpdivcl 11330 . . 3  |-  ( ( ( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR  /\  n  e.  RR+ )  -> 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n )  e.  RR )
1412, 13sylan 473 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n
)  e.  RR )
15 simpr 462 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  RR+ )
16 simpl 458 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
17 1red 9658 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
18 0lt1 10136 . . . . . . . 8  |-  0  <  1
1918a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
20 simpr 462 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
211, 17, 16, 19, 20lttrd 9796 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
2216, 21elrpd 11338 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
23 rpre 11308 . . . . 5  |-  ( n  e.  RR+  ->  n  e.  RR )
24 rpcxpcl 23607 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR )  ->  ( A  ^c  n )  e.  RR+ )
2522, 23, 24syl2an 479 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^c  n )  e.  RR+ )
2615, 25rpdivcld 11358 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  e.  RR+ )
2726rpred 11341 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  e.  RR )
283adantr 466 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( log `  A
)  e.  RR+ )
2915, 28rpmulcld 11357 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR+ )
3029rpred 11341 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR )
3130resqcld 12441 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  e.  RR )
3231rehalfcld 10859 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  e.  RR )
33 1rp 11306 . . . . . . . . . . 11  |-  1  e.  RR+
34 rpaddcl 11323 . . . . . . . . . . 11  |-  ( ( 1  e.  RR+  /\  (
n  x.  ( log `  A ) )  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A
) ) )  e.  RR+ )
3533, 29, 34sylancr 667 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR+ )
3635rpred 11341 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR )
3736, 32readdcld 9670 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )  e.  RR )
3830reefcld 14129 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( exp `  (
n  x.  ( log `  A ) ) )  e.  RR )
3932, 35ltaddrp2d 11372 . . . . . . . 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 ) ) )
40 efgt1p2 14155 . . . . . . . . 9  |-  ( ( n  x.  ( log `  A ) )  e.  RR+  ->  ( ( 1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )  <  ( exp `  ( n  x.  ( log `  A ) ) ) )
4129, 40syl 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 ) ) ) )
4232, 37, 38, 39, 41lttrd 9796 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  <  ( exp `  ( n  x.  ( log `  A ) ) ) )
4323adantl 467 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  RR )
4443recnd 9669 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  CC )
4544sqcld 12413 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  CC )
46 2cnd 10682 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  e.  CC )
476adantr 466 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  RR+ )
4847rpcnd 11343 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  CC )
49 2ne0 10702 . . . . . . . . . 10  |-  2  =/=  0
5049a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  =/=  0
)
5147rpne0d 11346 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  =/=  0 )
5245, 46, 48, 50, 51divdiv2d 10415 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n ^
2 )  x.  (
( log `  A
) ^ 2 ) )  /  2 ) )
533rpcnd 11343 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  CC )
5453adantr 466 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( log `  A
)  e.  CC )
5544, 54sqmuld 12427 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  =  ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) ) )
5655oveq1d 6316 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  =  ( ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) )  /  2 ) )
5752, 56eqtr4d 2466 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )
5816recnd 9669 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  CC )
5958adantr 466 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  e.  CC )
6022adantr 466 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  e.  RR+ )
6160rpne0d 11346 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  =/=  0
)
6259, 61, 44cxpefd 23643 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^c  n )  =  ( exp `  (
n  x.  ( log `  A ) ) ) )
6342, 57, 623brtr4d 4451 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  <  ( A  ^c  n ) )
64 rpexpcl 12290 . . . . . . . . 9  |-  ( ( n  e.  RR+  /\  2  e.  ZZ )  ->  (
n ^ 2 )  e.  RR+ )
6515, 4, 64sylancl 666 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  RR+ )
668adantr 466 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  e.  RR+ )
6765, 66rpdivcld 11358 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  e.  RR+ )
6867, 25, 15ltdiv2d 11364 . . . . . 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 ) ) ) ) ) )
6963, 68mpbid 213 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  < 
( n  /  (
( n ^ 2 )  /  ( 2  /  ( ( log `  A ) ^ 2 ) ) ) ) )
709adantr 466 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  e.  CC )
7165rpne0d 11346 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =/=  0
)
7266rpne0d 11346 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  =/=  0 )
7344, 45, 70, 71, 72divdiv2d 10415 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( ( n ^
2 )  /  (
2  /  ( ( log `  A ) ^ 2 ) ) ) )  =  ( ( n  x.  (
2  /  ( ( log `  A ) ^ 2 ) ) )  /  ( n ^ 2 ) ) )
7444sqvald 12412 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =  ( n  x.  n ) )
7574oveq2d 6317 . . . . . 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
) ) )
76 rpne0 11317 . . . . . . . 8  |-  ( n  e.  RR+  ->  n  =/=  0 )
7776adantl 467 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  =/=  0
)
7870, 44, 44, 77, 77divcan5d 10409 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( 2  / 
( ( log `  A
) ^ 2 ) ) )  /  (
n  x.  n ) )  =  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )
7973, 75, 783eqtrd 2467 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( ( n ^
2 )  /  (
2  /  ( ( log `  A ) ^ 2 ) ) ) )  =  ( ( 2  /  (
( log `  A
) ^ 2 ) )  /  n ) )
8069, 79breqtrd 4445 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  < 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8127, 14, 80ltled 9783 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  <_ 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8281adantrr 721 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( n  e.  RR+  /\  0  <_  n
) )  ->  (
n  /  ( A  ^c  n ) )  <_  ( (
2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )
8326rpge0d 11345 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  0  <_  (
n  /  ( A  ^c  n ) ) )
8483adantrr 721 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( n  e.  RR+  /\  0  <_  n
) )  ->  0  <_  ( n  /  ( A  ^c  n ) ) )
851, 1, 11, 14, 27, 82, 84rlimsqz2 13701 1  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( n  e.  RR+  |->  ( n  /  ( A  ^c  n ) ) )  ~~> r  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1868    =/= wne 2618   class class class wbr 4420    |-> cmpt 4479   ` cfv 5597  (class class class)co 6301   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    / cdiv 10269   2c2 10659   ZZcz 10937   RR+crp 11302   ^cexp 12271    ~~> r crli 13536   expce 14101   logclog 23490    ^c ccxp 23491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-shft 13118  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-limsup 13513  df-clim 13539  df-rlim 13540  df-sum 13740  df-ef 14108  df-sin 14110  df-cos 14111  df-pi 14113  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cncf 21896  df-limc 22807  df-dv 22808  df-log 23492  df-cxp 23493
This theorem is referenced by:  cxp2lim  23888  cxploglim  23889
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