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Theorem cxplim 20262
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem cxplim
StepHypRef Expression
1 rpre 10357 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  RR )
21adantl 454 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  x  e.  RR )
3 rpge0 10363 . . . . . 6  |-  ( x  e.  RR+  ->  0  <_  x )
43adantl 454 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  0  <_  x )
5 rpre 10357 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  RR )
65renegcld 9207 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  e.  RR )
76adantr 453 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  e.  RR )
8 rpcn 10359 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  CC )
9 rpne0 10366 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
108, 9negne0d 9152 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  =/=  0 )
1110adantr 453 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  =/=  0 )
127, 11rereccld 9584 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
1  /  -u A
)  e.  RR )
132, 4, 12recxpcld 20066 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
x  ^ c  ( 1  /  -u A
) )  e.  RR )
14 simprl 734 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  RR+ )
155ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  RR )
16 rpcxpcl 20019 . . . . . . . . . . 11  |-  ( ( n  e.  RR+  /\  A  e.  RR )  ->  (
n  ^ c  A
)  e.  RR+ )
1714, 15, 16syl2anc 644 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  A
)  e.  RR+ )
1817rpreccld 10397 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^ c  A ) )  e.  RR+ )
1918rprege0d 10394 . . . . . . . 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 11777 . . . . . . . 8  |-  ( ( ( 1  /  (
n  ^ c  A
) )  e.  RR  /\  0  <_  ( 1  /  ( n  ^ c  A ) ) )  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  =  ( 1  /  ( n  ^ c  A ) ) )
2119, 20syl 17 . . . . . . 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 733 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  RR+ )
23 simprr 735 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
x  ^ c  ( 1  /  -u A
) )  <  n
)
24 rpreccl 10374 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
2524ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  RR+ )
2625rpcnd 10389 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  CC )
2722, 26cxprecd 20072 . . . . . . . . . . 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 10359 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  CC )
2928ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  CC )
30 rpne0 10366 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  =/=  0 )
3130ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  =/=  0 )
3229, 31, 26cxpnegd 20058 . . . . . . . . . . 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 8792 . . . . . . . . . . . . . 14  |-  1  e.  CC
3433a1i 12 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  1  e.  CC )
358ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  CC )
369ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  =/=  0 )
3734, 35, 36divneg2d 9547 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
3837oveq2d 5837 . . . . . . . . . . 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 2324 . . . . . . . . . 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 9528 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
4140oveq2d 5837 . . . . . . . . . . 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 20077 . . . . . . . . . . 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 10389 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  CC )
4443cxp1d 20049 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  1 )  =  n )
4541, 42, 443eqtr3d 2326 . . . . . . . . . 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 4056 . . . . . . . . 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 10374 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
4847ad2antlr 709 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR+ )
4948rpred 10387 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR )
5048rpge0d 10391 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( 1  /  x
) )
5117rpred 10387 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  A
)  e.  RR )
5217rpge0d 10391 . . . . . . . . . 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 20069 . . . . . . . . 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 225 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  <  ( n  ^ c  A ) )
5522, 17, 54ltrec1d 10407 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^ c  A ) )  <  x )
5621, 55eqbrtrd 4046 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( abs `  ( 1  / 
( n  ^ c  A ) ) )  <  x )
5756expr 600 . . . . 5  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( x  ^ c  ( 1  /  -u A ) )  < 
n  ->  ( abs `  ( 1  /  (
n  ^ c  A
) ) )  < 
x ) )
5857ralrimiva 2629 . . . 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 4029 . . . . . . 7  |-  ( y  =  ( x  ^ c  ( 1  /  -u A ) )  -> 
( y  <  n  <->  ( x  ^ c  ( 1  /  -u A
) )  <  n
) )
6059imbi1d 310 . . . . . 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 2566 . . . . 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 2887 . . . 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 644 . . 3  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
) )
6463ralrimiva 2629 . 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 21 . . . . . . 7  |-  ( n  e.  RR+  ->  n  e.  RR+ )
6665, 5, 16syl2anr 466 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^ c  A
)  e.  RR+ )
6766rpreccld 10397 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^ c  A ) )  e.  RR+ )
6867rpcnd 10389 . . . 4  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^ c  A ) )  e.  CC )
6968ralrimiva 2629 . . 3  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( 1  / 
( n  ^ c  A ) )  e.  CC )
70 rpssre 10361 . . . 4  |-  RR+  C_  RR
7170a1i 12 . . 3  |-  ( A  e.  RR+  ->  RR+  C_  RR )
7269, 71rlim0lt 11979 . 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 225 1  |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( 1  /  ( n  ^ c  A ) ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1625    e. wcel 1687    =/= wne 2449   A.wral 2546   E.wrex 2547    C_ wss 3155   class class class wbr 4026    e. cmpt 4080   ` cfv 5223  (class class class)co 5821   CCcc 8732   RRcr 8733   0cc0 8734   1c1 8735    x. cmul 8739    < clt 8864    <_ cle 8865   -ucneg 9035    / cdiv 9420   RR+crp 10351   abscabs 11715    ~~> r crli 11955    ^ c ccxp 19909
This theorem is referenced by:  sqrlim  20263
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812  ax-addf 8813  ax-mulf 8814
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-of 6041  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-er 6657  df-map 6771  df-pm 6772  df-ixp 6815  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-fi 7162  df-sup 7191  df-oi 7222  df-card 7569  df-cda 7791  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-4 9803  df-5 9804  df-6 9805  df-7 9806  df-8 9807  df-9 9808  df-10 9809  df-n0 9963  df-z 10022  df-dec 10122  df-uz 10228  df-q 10314  df-rp 10352  df-xneg 10449  df-xadd 10450  df-xmul 10451  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-sum 12155  df-ef 12345  df-sin 12347  df-cos 12348  df-pi 12350  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-limc 19212  df-dv 19213  df-log 19910  df-cxp 19911
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