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Theorem cxplim 20677
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 10550 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  RR )
21adantl 453 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  x  e.  RR )
3 rpge0 10556 . . . . . 6  |-  ( x  e.  RR+  ->  0  <_  x )
43adantl 453 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  0  <_  x )
5 rpre 10550 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  RR )
65renegcld 9396 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  e.  RR )
76adantr 452 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  e.  RR )
8 rpcn 10552 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  CC )
9 rpne0 10559 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
108, 9negne0d 9341 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  =/=  0 )
1110adantr 452 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  =/=  0 )
127, 11rereccld 9773 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
1  /  -u A
)  e.  RR )
132, 4, 12recxpcld 20481 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
x  ^ c  ( 1  /  -u A
) )  e.  RR )
14 simprl 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  RR+ )
155ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  RR )
1614, 15rpcxpcld 20488 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  A
)  e.  RR+ )
1716rpreccld 10590 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^ c  A ) )  e.  RR+ )
1817rprege0d 10587 . . . . . . . 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 ) ) ) )
19 absid 12028 . . . . . . . 8  |-  ( ( ( 1  /  (
n  ^ c  A
) )  e.  RR  /\  0  <_  ( 1  /  ( n  ^ c  A ) ) )  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  =  ( 1  /  ( n  ^ c  A ) ) )
2018, 19syl 16 . . . . . . 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 ) ) )
21 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  RR+ )
22 simprr 734 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
x  ^ c  ( 1  /  -u A
) )  <  n
)
23 rpreccl 10567 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
2423ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  RR+ )
2524rpcnd 10582 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  CC )
2621, 25cxprecd 20487 . . . . . . . . . . 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 ) ) ) )
27 rpcn 10552 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  CC )
2827ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  CC )
29 rpne0 10559 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  =/=  0 )
3029ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  =/=  0 )
3128, 30, 25cxpnegd 20473 . . . . . . . . . . 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 ) ) ) )
32 ax-1cn 8981 . . . . . . . . . . . . . 14  |-  1  e.  CC
3332a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  1  e.  CC )
348ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  CC )
359ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  =/=  0 )
3633, 34, 35divneg2d 9736 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
3736oveq2d 6036 . . . . . . . . . . 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
) ) )
3826, 31, 373eqtr2d 2425 . . . . . . . . . 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 ) ) )
3934, 35recidd 9717 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
4039oveq2d 6036 . . . . . . . . . . 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 ) )
4114, 15, 25cxpmuld 20492 . . . . . . . . . . 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 ) ) )
4214rpcnd 10582 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  CC )
4342cxp1d 20464 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  1 )  =  n )
4440, 41, 433eqtr3d 2427 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
( n  ^ c  A )  ^ c 
( 1  /  A
) )  =  n )
4522, 38, 443brtr4d 4183 . . . . . . . . 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 ) ) )
46 rpreccl 10567 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
4746ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR+ )
4847rpred 10580 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR )
4947rpge0d 10584 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( 1  /  x
) )
5016rpred 10580 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  A
)  e.  RR )
5116rpge0d 10584 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( n  ^ c  A ) )
5248, 49, 50, 51, 24cxplt2d 20484 . . . . . . . . 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 ) ) ) )
5345, 52mpbird 224 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  <  ( n  ^ c  A ) )
5421, 16, 53ltrec1d 10600 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^ c  A ) )  <  x )
5520, 54eqbrtrd 4173 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( abs `  ( 1  / 
( n  ^ c  A ) ) )  <  x )
5655expr 599 . . . . 5  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( x  ^ c  ( 1  /  -u A ) )  < 
n  ->  ( abs `  ( 1  /  (
n  ^ c  A
) ) )  < 
x ) )
5756ralrimiva 2732 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  A. n  e.  RR+  ( ( x  ^ c  ( 1  /  -u A ) )  <  n  ->  ( abs `  ( 1  / 
( n  ^ c  A ) ) )  <  x ) )
58 breq1 4156 . . . . . . 7  |-  ( y  =  ( x  ^ c  ( 1  /  -u A ) )  -> 
( y  <  n  <->  ( x  ^ c  ( 1  /  -u A
) )  <  n
) )
5958imbi1d 309 . . . . . 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
) ) )
6059ralbidv 2669 . . . . 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 ) ) )
6160rspcev 2995 . . . 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 ) )
6213, 57, 61syl2anc 643 . . 3  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
) )
6362ralrimiva 2732 . 2  |-  ( A  e.  RR+  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
) )
64 id 20 . . . . . . 7  |-  ( n  e.  RR+  ->  n  e.  RR+ )
65 rpcxpcl 20434 . . . . . . 7  |-  ( ( n  e.  RR+  /\  A  e.  RR )  ->  (
n  ^ c  A
)  e.  RR+ )
6664, 5, 65syl2anr 465 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^ c  A
)  e.  RR+ )
6766rpreccld 10590 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^ c  A ) )  e.  RR+ )
6867rpcnd 10582 . . . 4  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^ c  A ) )  e.  CC )
6968ralrimiva 2732 . . 3  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( 1  / 
( n  ^ c  A ) )  e.  CC )
70 rpssre 10554 . . . 4  |-  RR+  C_  RR
7170a1i 11 . . 3  |-  ( A  e.  RR+  ->  RR+  C_  RR )
7269, 71rlim0lt 12230 . 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
) ) )
7363, 72mpbird 224 1  |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( 1  /  ( n  ^ c  A ) ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650    C_ wss 3263   class class class wbr 4153    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928    < clt 9053    <_ cle 9054   -ucneg 9224    / cdiv 9609   RR+crp 10544   abscabs 11966    ~~> r crli 12206    ^ c ccxp 20320
This theorem is referenced by:  sqrlim  20678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-sum 12407  df-ef 12597  df-sin 12599  df-cos 12600  df-pi 12602  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-limc 19620  df-dv 19621  df-log 20321  df-cxp 20322
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