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Theorem cxplim 20228
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
StepHypRef Expression
1 rpre 10327 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  RR )
21adantl 454 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  x  e.  RR )
3 rpge0 10333 . . . . . 6  |-  ( x  e.  RR+  ->  0  <_  x )
43adantl 454 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  0  <_  x )
5 rpre 10327 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  RR )
65renegcld 9178 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  e.  RR )
76adantr 453 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  e.  RR )
8 rpcn 10329 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  CC )
9 rpne0 10336 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
108, 9negne0d 9123 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  =/=  0 )
1110adantr 453 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  =/=  0 )
127, 11rereccld 9555 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
1  /  -u A
)  e.  RR )
132, 4, 12recxpcld 20032 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
x  ^ c  ( 1  /  -u A
) )  e.  RR )
14 simprl 735 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  RR+ )
155ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  RR )
16 rpcxpcl 19985 . . . . . . . . . . 11  |-  ( ( n  e.  RR+  /\  A  e.  RR )  ->  (
n  ^ c  A
)  e.  RR+ )
1714, 15, 16syl2anc 645 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  A
)  e.  RR+ )
1817rpreccld 10367 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^ c  A ) )  e.  RR+ )
1918rprege0d 10364 . . . . . . . 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 11746 . . . . . . . 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 734 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  RR+ )
23 simprr 736 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
x  ^ c  ( 1  /  -u A
) )  <  n
)
24 rpreccl 10344 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
2524ad2antrr 709 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  RR+ )
2625rpcnd 10359 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  CC )
2722, 26cxprecd 20038 . . . . . . . . . . 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 10329 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  CC )
2928ad2antlr 710 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  CC )
30 rpne0 10336 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  =/=  0 )
3130ad2antlr 710 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  x  =/=  0 )
3229, 31, 26cxpnegd 20024 . . . . . . . . . . 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 8763 . . . . . . . . . . . . . 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 709 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  CC )
369ad2antrr 709 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  A  =/=  0 )
3734, 35, 36divneg2d 9518 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
3837oveq2d 5808 . . . . . . . . . . 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 2296 . . . . . . . . . 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 9499 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
4140oveq2d 5808 . . . . . . . . . . 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 20043 . . . . . . . . . . 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 10359 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  CC )
4443cxp1d 20015 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  1 )  =  n )
4541, 42, 443eqtr3d 2298 . . . . . . . . . 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 4027 . . . . . . . . 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 10344 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
4847ad2antlr 710 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR+ )
4948rpred 10357 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR )
5048rpge0d 10361 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( 1  /  x
) )
5117rpred 10357 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^ c  A
)  e.  RR )
5217rpge0d 10361 . . . . . . . . . 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 20035 . . . . . . . . 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 10377 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^ c  A ) )  <  x )
5621, 55eqbrtrd 4017 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^ c 
( 1  /  -u A
) )  <  n
) )  ->  ( abs `  ( 1  / 
( n  ^ c  A ) ) )  <  x )
5756expr 601 . . . . 5  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( x  ^ c  ( 1  /  -u A ) )  < 
n  ->  ( abs `  ( 1  /  (
n  ^ c  A
) ) )  < 
x ) )
5857ralrimiva 2601 . . . 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 4000 . . . . . . 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 2538 . . . . 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 ) ) )
6261rcla4ev 2859 . . . 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 645 . . 3  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^ c  A ) ) )  <  x
) )
6463ralrimiva 2601 . 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 10367 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^ c  A ) )  e.  RR+ )
6867rpcnd 10359 . . . 4  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^ c  A ) )  e.  CC )
6968ralrimiva 2601 . . 3  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( 1  / 
( n  ^ c  A ) )  e.  CC )
70 rpssre 10331 . . . 4  |-  RR+  C_  RR
7170a1i 12 . . 3  |-  ( A  e.  RR+  ->  RR+  C_  RR )
7269, 71rlim0lt 11948 . 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 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   E.wrex 2519    C_ wss 3127   class class class wbr 3997    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    x. cmul 8710    < clt 8835    <_ cle 8836   -ucneg 9006    / cdiv 9391   RR+crp 10321   abscabs 11684    ~~> r crli 11924    ^ c ccxp 19875
This theorem is referenced by:  sqrlim  20229
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-q 10284  df-rp 10322  df-xneg 10419  df-xadd 10420  df-xmul 10421  df-ioo 10626  df-ioc 10627  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-fl 10891  df-mod 10940  df-seq 11013  df-exp 11071  df-fac 11255  df-bc 11282  df-hash 11304  df-shft 11527  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-limsup 11910  df-clim 11927  df-rlim 11928  df-sum 12124  df-ef 12311  df-sin 12313  df-cos 12314  df-pi 12316  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-starv 13185  df-sca 13186  df-vsca 13187  df-tset 13189  df-ple 13190  df-ds 13192  df-hom 13194  df-cco 13195  df-rest 13289  df-topn 13290  df-topgen 13306  df-pt 13307  df-prds 13310  df-xrs 13365  df-0g 13366  df-gsum 13367  df-qtop 13372  df-imas 13373  df-xps 13375  df-mre 13450  df-mrc 13451  df-acs 13453  df-mnd 14329  df-submnd 14378  df-mulg 14454  df-cntz 14755  df-cmn 15053  df-xmet 16335  df-met 16336  df-bl 16337  df-mopn 16338  df-cnfld 16340  df-top 16598  df-bases 16600  df-topon 16601  df-topsp 16602  df-cld 16718  df-ntr 16719  df-cls 16720  df-nei 16797  df-lp 16830  df-perf 16831  df-cn 16919  df-cnp 16920  df-haus 17005  df-tx 17219  df-hmeo 17408  df-fbas 17482  df-fg 17483  df-fil 17503  df-fm 17595  df-flim 17596  df-flf 17597  df-xms 17847  df-ms 17848  df-tms 17849  df-cncf 18344  df-limc 19178  df-dv 19179  df-log 19876  df-cxp 19877
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