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Theorem cxplim 20266
Description: A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)
Assertion
Ref Expression
cxplim  |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( 1  /  ( n  ^ c  A ) ) )  ~~> r  0 )
Distinct variable group:    A, n

Proof of Theorem cxplim
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rpre 10360 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  RR )
21adantl 452 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  x  e.  RR )
3 rpge0 10366 . . . . . 6  |-  ( x  e.  RR+  ->  0  <_  x )
43adantl 452 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  0  <_  x )
5 rpre 10360 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  RR )
65renegcld 9210 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  e.  RR )
76adantr 451 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  e.  RR )
8 rpcn 10362 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  CC )
9 rpne0 10369 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
108, 9negne0d 9155 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  =/=  0 )
1110adantr 451 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  =/=  0 )
127, 11rereccld 9587 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
1  /  -u A
)  e.  RR )
132, 4, 12recxpcld 20070 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
x  ^ c  ( 1  /  -u A
) )  e.  RR )
14 simprl 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  RR+ )
155ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  RR )
16 rpcxpcl 20023 . . . . . . . . . . 11  |-  ( ( n  e.  RR+  /\  A  e.  RR )  ->  (
n  ^ c  A
)  e.  RR+ )
1714, 15, 16syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  A
)  e.  RR+ )
1817rpreccld 10400 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^ c  A ) )  e.  RR+ )
1918rprege0d 10397 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
( 1  /  (
n  ^ c  A
) )  e.  RR  /\  0  <_  ( 1  /  ( n  ^ c  A ) ) ) )
20 absid 11781 . . . . . . . 8  |-  ( ( ( 1  /  (
n  ^ c  A
) )  e.  RR  /\  0  <_  ( 1  /  ( n  ^ c  A ) ) )  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  =  ( 1  /  ( n  ^ c  A ) ) )
2119, 20syl 15 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( abs `  ( 1  / 
( n  ^ c  A ) ) )  =  ( 1  / 
( n  ^ c  A ) ) )
22 simplr 731 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  RR+ )
23 simprr 733 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
x  ^ c  ( 1  /  -u A
) )  <  n
)
24 rpreccl 10377 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
2524ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  RR+ )
2625rpcnd 10392 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  CC )
2722, 26cxprecd 20076 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
( 1  /  x
)  ^ c  ( 1  /  A ) )  =  ( 1  /  ( x  ^ c  ( 1  /  A ) ) ) )
28 rpcn 10362 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  CC )
2928ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  CC )
30 rpne0 10369 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  =/=  0 )
3130ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  =/=  0 )
3229, 31, 26cxpnegd 20062 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
x  ^ c  -u ( 1  /  A
) )  =  ( 1  /  ( x  ^ c  ( 1  /  A ) ) ) )
33 ax-1cn 8795 . . . . . . . . . . . . . 14  |-  1  e.  CC
3433a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  1  e.  CC )
358ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  CC )
369ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  =/=  0 )
3734, 35, 36divneg2d 9550 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
3837oveq2d 5874 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
x  ^ c  -u ( 1  /  A
) )  =  ( x  ^ c  ( 1  /  -u A
) ) )
3927, 32, 383eqtr2d 2321 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
( 1  /  x
)  ^ c  ( 1  /  A ) )  =  ( x  ^ c  ( 1  /  -u A ) ) )
4035, 36recidd 9531 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
4140oveq2d 5874 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  ( A  x.  ( 1  /  A ) ) )  =  ( n  ^ c  1 ) )
4214, 15, 26cxpmuld 20081 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  ( A  x.  ( 1  /  A ) ) )  =  ( ( n  ^ c  A
)  ^ c  ( 1  /  A ) ) )
4314rpcnd 10392 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  CC )
4443cxp1d 20053 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  1 )  =  n )
4541, 42, 443eqtr3d 2323 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
( n  ^ c  A )  ^ c 
( 1  /  A
) )  =  n )
4623, 39, 453brtr4d 4053 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
( 1  /  x
)  ^ c  ( 1  /  A ) )  <  ( ( n  ^ c  A
)  ^ c  ( 1  /  A ) ) )
47 rpreccl 10377 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
4847ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR+ )
4948rpred 10390 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR )
5048rpge0d 10394 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( 1  /  x
) )
5117rpred 10390 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  A
)  e.  RR )
5217rpge0d 10394 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( n  ^ c  A ) )
5349, 50, 51, 52, 25cxplt2d 20073 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
( 1  /  x
)  <  ( n  ^ c  A )  <->  ( ( 1  /  x
)  ^ c  ( 1  /  A ) )  <  ( ( n  ^ c  A
)  ^ c  ( 1  /  A ) ) ) )
5446, 53mpbird 223 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  <  ( n  ^ c  A ) )
5522, 17, 54ltrec1d 10410 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^ c  A ) )  <  x )
5621, 55eqbrtrd 4043 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( abs `  ( 1  / 
( n  ^ c  A ) ) )  <  x )
5756expr 598 . . . . 5  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( x  ^ c  ( 1  /  -u A ) )  < 
n  ->  ( abs `  ( 1  /  (
n  ^ c  A
) ) )  < 
x ) )
5857ralrimiva 2626 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  A. n  e.  RR+  ( ( x  ^ c  ( 1  /  -u A ) )  <  n  ->  ( abs `  ( 1  / 
( n  ^ c  A ) ) )  <  x ) )
59 breq1 4026 . . . . . . 7  |-  ( y  =  ( x  ^ c  ( 1  /  -u A ) )  -> 
( y  <  n  <->  ( x  ^ c  ( 1  /  -u A
) )  <  n
) )
6059imbi1d 308 . . . . . 6  |-  ( y  =  ( x  ^ c  ( 1  /  -u A ) )  -> 
( ( y  < 
n  ->  ( abs `  ( 1  /  (
n  ^ c  A
) ) )  < 
x )  <->  ( (
x  ^ c  ( 1  /  -u A
) )  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
) ) )
6160ralbidv 2563 . . . . 5  |-  ( y  =  ( x  ^ c  ( 1  /  -u A ) )  -> 
( A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
)  <->  A. n  e.  RR+  ( ( x  ^ c  ( 1  /  -u A ) )  < 
n  ->  ( abs `  ( 1  /  (
n  ^ c  A
) ) )  < 
x ) ) )
6261rspcev 2884 . . . 4  |-  ( ( ( x  ^ c 
( 1  /  -u A
) )  e.  RR  /\ 
A. n  e.  RR+  ( ( x  ^ c  ( 1  /  -u A ) )  < 
n  ->  ( abs `  ( 1  /  (
n  ^ c  A
) ) )  < 
x ) )  ->  E. y  e.  RR  A. n  e.  RR+  (
y  <  n  ->  ( abs `  ( 1  /  ( n  ^ c  A ) ) )  <  x ) )
6313, 58, 62syl2anc 642 . . 3  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
) )
6463ralrimiva 2626 . 2  |-  ( A  e.  RR+  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
) )
65 id 19 . . . . . . 7  |-  ( n  e.  RR+  ->  n  e.  RR+ )
6665, 5, 16syl2anr 464 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^ c  A
)  e.  RR+ )
6766rpreccld 10400 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^ c  A ) )  e.  RR+ )
6867rpcnd 10392 . . . 4  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^ c  A ) )  e.  CC )
6968ralrimiva 2626 . . 3  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( 1  / 
( n  ^ c  A ) )  e.  CC )
70 rpssre 10364 . . . 4  |-  RR+  C_  RR
7170a1i 10 . . 3  |-  ( A  e.  RR+  ->  RR+  C_  RR )
7269, 71rlim0lt 11983 . 2  |-  ( A  e.  RR+  ->  ( ( n  e.  RR+  |->  ( 1  /  ( n  ^ c  A ) ) )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
) ) )
7364, 72mpbird 223 1  |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( 1  /  ( n  ^ c  A ) ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868   -ucneg 9038    / cdiv 9423   RR+crp 10354   abscabs 11719    ~~> r crli 11959    ^ c ccxp 19913
This theorem is referenced by:  sqrlim  20267
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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