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Theorem cxp2limlem 22254
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 9375 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
2 2rp 10984 . . . . 5  |-  2  e.  RR+
3 rplogcl 21938 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
4 2z 10666 . . . . . 6  |-  2  e.  ZZ
5 rpexpcl 11868 . . . . . 6  |-  ( ( ( log `  A
)  e.  RR+  /\  2  e.  ZZ )  ->  (
( log `  A
) ^ 2 )  e.  RR+ )
63, 4, 5sylancl 655 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( ( log `  A
) ^ 2 )  e.  RR+ )
7 rpdivcl 11001 . . . . 5  |-  ( ( 2  e.  RR+  /\  (
( log `  A
) ^ 2 )  e.  RR+ )  ->  (
2  /  ( ( log `  A ) ^ 2 ) )  e.  RR+ )
82, 6, 7sylancr 656 . . . 4  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR+ )
98rpcnd 11017 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  CC )
10 divrcnv 13298 . . 3  |-  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  e.  CC  ->  (
n  e.  RR+  |->  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )  ~~> r  0 )
119, 10syl 16 . 2  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( n  e.  RR+  |->  ( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )  ~~> r  0 )
128rpred 11015 . . 3  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR )
13 rerpdivcl 11006 . . 3  |-  ( ( ( 2  /  (
( log `  A
) ^ 2 ) )  e.  RR  /\  n  e.  RR+ )  -> 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n )  e.  RR )
1412, 13sylan 468 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 2  /  ( ( log `  A ) ^ 2 ) )  /  n
)  e.  RR )
15 simpr 458 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  RR+ )
16 simpl 454 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
17 1red 9389 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
18 0lt1 9850 . . . . . . . 8  |-  0  <  1
1918a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
20 simpr 458 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
211, 17, 16, 19, 20lttrd 9520 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
2216, 21elrpd 11013 . . . . 5  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
23 rpre 10985 . . . . 5  |-  ( n  e.  RR+  ->  n  e.  RR )
24 rpcxpcl 22006 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR )  ->  ( A  ^c  n )  e.  RR+ )
2522, 23, 24syl2an 474 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^c  n )  e.  RR+ )
2615, 25rpdivcld 11032 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  e.  RR+ )
2726rpred 11015 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  e.  RR )
283adantr 462 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( log `  A
)  e.  RR+ )
2915, 28rpmulcld 11031 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR+ )
3029rpred 11015 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  x.  ( log `  A
) )  e.  RR )
3130resqcld 12018 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  e.  RR )
3231rehalfcld 10559 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  e.  RR )
33 1rp 10983 . . . . . . . . . . 11  |-  1  e.  RR+
34 rpaddcl 10999 . . . . . . . . . . 11  |-  ( ( 1  e.  RR+  /\  (
n  x.  ( log `  A ) )  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A
) ) )  e.  RR+ )
3533, 29, 34sylancr 656 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR+ )
3635rpred 11015 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 1  +  ( n  x.  ( log `  A ) ) )  e.  RR )
3736, 32readdcld 9401 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( 1  +  ( n  x.  ( log `  A
) ) )  +  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )  e.  RR )
3830reefcld 13356 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( exp `  (
n  x.  ( log `  A ) ) )  e.  RR )
3932, 35ltaddrp2d 11045 . . . . . . . 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 13381 . . . . . . . . 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 16 . . . . . . . 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 9520 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  <  ( exp `  ( n  x.  ( log `  A ) ) ) )
4323adantl 463 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  RR )
4443recnd 9400 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  e.  CC )
4544sqcld 11990 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  CC )
46 2cnd 10382 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  e.  CC )
476adantr 462 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  RR+ )
4847rpcnd 11017 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  e.  CC )
49 2ne0 10402 . . . . . . . . . 10  |-  2  =/=  0
5049a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  2  =/=  0
)
5147rpne0d 11020 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( log `  A ) ^ 2 )  =/=  0 )
5245, 46, 48, 50, 51divdiv2d 10127 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n ^
2 )  x.  (
( log `  A
) ^ 2 ) )  /  2 ) )
533rpcnd 11017 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  CC )
5453adantr 462 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( log `  A
)  e.  CC )
5544, 54sqmuld 12004 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n  x.  ( log `  A
) ) ^ 2 )  =  ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) ) )
5655oveq1d 6095 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( ( n  x.  ( log `  A ) ) ^
2 )  /  2
)  =  ( ( ( n ^ 2 )  x.  ( ( log `  A ) ^ 2 ) )  /  2 ) )
5752, 56eqtr4d 2468 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  =  ( ( ( n  x.  ( log `  A
) ) ^ 2 )  /  2 ) )
5816recnd 9400 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  CC )
5958adantr 462 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  e.  CC )
6022adantr 462 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  e.  RR+ )
6160rpne0d 11020 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  A  =/=  0
)
6259, 61, 44cxpefd 22042 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( A  ^c  n )  =  ( exp `  (
n  x.  ( log `  A ) ) ) )
6342, 57, 623brtr4d 4310 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  <  ( A  ^c  n ) )
64 rpexpcl 11868 . . . . . . . . 9  |-  ( ( n  e.  RR+  /\  2  e.  ZZ )  ->  (
n ^ 2 )  e.  RR+ )
6515, 4, 64sylancl 655 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  e.  RR+ )
668adantr 462 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  e.  RR+ )
6765, 66rpdivcld 11032 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( ( n ^ 2 )  / 
( 2  /  (
( log `  A
) ^ 2 ) ) )  e.  RR+ )
6867, 25, 15ltdiv2d 11038 . . . . . 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 210 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  < 
( n  /  (
( n ^ 2 )  /  ( 2  /  ( ( log `  A ) ^ 2 ) ) ) ) )
709adantr 462 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  e.  CC )
7165rpne0d 11020 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =/=  0
)
7266rpne0d 11020 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( 2  / 
( ( log `  A
) ^ 2 ) )  =/=  0 )
7344, 45, 70, 71, 72divdiv2d 10127 . . . . . 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 11989 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n ^
2 )  =  ( n  x.  n ) )
7574oveq2d 6096 . . . . . 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 10994 . . . . . . . 8  |-  ( n  e.  RR+  ->  n  =/=  0 )
7776adantl 463 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  n  =/=  0
)
7870, 44, 44, 77, 77divcan5d 10121 . . . . . 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 2469 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( ( n ^
2 )  /  (
2  /  ( ( log `  A ) ^ 2 ) ) ) )  =  ( ( 2  /  (
( log `  A
) ^ 2 ) )  /  n ) )
8069, 79breqtrd 4304 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  < 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8127, 14, 80ltled 9510 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  ( n  / 
( A  ^c 
n ) )  <_ 
( ( 2  / 
( ( log `  A
) ^ 2 ) )  /  n ) )
8281adantrr 709 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( n  e.  RR+  /\  0  <_  n
) )  ->  (
n  /  ( A  ^c  n ) )  <_  ( (
2  /  ( ( log `  A ) ^ 2 ) )  /  n ) )
8326rpge0d 11019 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  n  e.  RR+ )  ->  0  <_  (
n  /  ( A  ^c  n ) ) )
8483adantrr 709 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( n  e.  RR+  /\  0  <_  n
) )  ->  0  <_  ( n  /  ( A  ^c  n ) ) )
851, 1, 11, 14, 27, 82, 84rlimsqz2 13112 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 369    e. wcel 1755    =/= wne 2596   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    <_ cle 9407    / cdiv 9981   2c2 10359   ZZcz 10634   RR+crp 10979   ^cexp 11849    ~~> r crli 12947   expce 13330   logclog 21891    ^c ccxp 21892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894
This theorem is referenced by:  cxp2lim  22255  cxploglim  22256
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