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Theorem cxplim 22308
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 10993 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  RR )
21adantl 463 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  x  e.  RR )
3 rpge0 10999 . . . . . 6  |-  ( x  e.  RR+  ->  0  <_  x )
43adantl 463 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  0  <_  x )
5 rpre 10993 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  RR )
65renegcld 9771 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  e.  RR )
76adantr 462 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  e.  RR )
8 rpcn 10995 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  CC )
9 rpne0 11002 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
108, 9negne0d 9713 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  =/=  0 )
1110adantr 462 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  =/=  0 )
127, 11rereccld 10154 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
1  /  -u A
)  e.  RR )
132, 4, 12recxpcld 22111 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
x  ^c  ( 1  /  -u A
) )  e.  RR )
14 simprl 750 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  RR+ )
155ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  RR )
1614, 15rpcxpcld 22118 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  A )  e.  RR+ )
1716rpreccld 11033 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^c  A ) )  e.  RR+ )
1817rprege0d 11030 . . . . . . . 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 12781 . . . . . . . 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 749 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  RR+ )
22 simprr 751 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
x  ^c  ( 1  /  -u A
) )  <  n
)
23 rpreccl 11010 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
2423ad2antrr 720 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  RR+ )
2524rpcnd 11025 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  CC )
2621, 25cxprecd 22117 . . . . . . . . . . 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 10995 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  CC )
2827ad2antlr 721 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  CC )
29 rpne0 11002 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  =/=  0 )
3029ad2antlr 721 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  x  =/=  0 )
3128, 30, 25cxpnegd 22103 . . . . . . . . . . 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 1cnd 9398 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  1  e.  CC )
338ad2antrr 720 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  CC )
349ad2antrr 720 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  A  =/=  0 )
3532, 33, 34divneg2d 10117 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
3635oveq2d 6106 . . . . . . . . . . 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
) ) )
3726, 31, 363eqtr2d 2479 . . . . . . . . . 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 ) ) )
3833, 34recidd 10098 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
3938oveq2d 6106 . . . . . . . . . . 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 ) )
4014, 15, 25cxpmuld 22122 . . . . . . . . . . 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 ) ) )
4114rpcnd 11025 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  CC )
4241cxp1d 22094 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  1 )  =  n )
4339, 40, 423eqtr3d 2481 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
( n  ^c  A )  ^c 
( 1  /  A
) )  =  n )
4422, 37, 433brtr4d 4319 . . . . . . . . 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 ) ) )
45 rpreccl 11010 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
4645ad2antlr 721 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR+ )
4746rpred 11023 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR )
4846rpge0d 11027 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( 1  /  x
) )
4916rpred 11023 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  A )  e.  RR )
5016rpge0d 11027 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( n  ^c  A ) )
5147, 48, 49, 50, 24cxplt2d 22114 . . . . . . . . 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 ) ) ) )
5244, 51mpbird 232 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  <  ( n  ^c  A ) )
5321, 16, 52ltrec1d 11043 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^c  A ) )  <  x )
5420, 53eqbrtrd 4309 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  ( abs `  ( 1  / 
( n  ^c  A ) ) )  <  x )
5554expr 612 . . . . 5  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( x  ^c  ( 1  /  -u A ) )  < 
n  ->  ( abs `  ( 1  /  (
n  ^c  A ) ) )  < 
x ) )
5655ralrimiva 2797 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  A. n  e.  RR+  ( ( x  ^c  ( 1  /  -u A ) )  <  n  ->  ( abs `  ( 1  / 
( n  ^c  A ) ) )  <  x ) )
57 breq1 4292 . . . . . . 7  |-  ( y  =  ( x  ^c  ( 1  /  -u A ) )  -> 
( y  <  n  <->  ( x  ^c  ( 1  /  -u A
) )  <  n
) )
5857imbi1d 317 . . . . . 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
) ) )
5958ralbidv 2733 . . . . 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 ) ) )
6059rspcev 3070 . . . 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 ) )
6113, 56, 60syl2anc 656 . . 3  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^c  A ) ) )  <  x
) )
6261ralrimiva 2797 . 2  |-  ( A  e.  RR+  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^c  A ) ) )  <  x
) )
63 id 22 . . . . . . 7  |-  ( n  e.  RR+  ->  n  e.  RR+ )
64 rpcxpcl 22064 . . . . . . 7  |-  ( ( n  e.  RR+  /\  A  e.  RR )  ->  (
n  ^c  A )  e.  RR+ )
6563, 5, 64syl2anr 475 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^c  A )  e.  RR+ )
6665rpreccld 11033 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^c  A ) )  e.  RR+ )
6766rpcnd 11025 . . . 4  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^c  A ) )  e.  CC )
6867ralrimiva 2797 . . 3  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( 1  / 
( n  ^c  A ) )  e.  CC )
69 rpssre 10997 . . . 4  |-  RR+  C_  RR
7069a1i 11 . . 3  |-  ( A  e.  RR+  ->  RR+  C_  RR )
7168, 70rlim0lt 12983 . 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
) ) )
7262, 71mpbird 232 1  |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( 1  /  ( n  ^c  A ) ) )  ~~> r  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714    C_ wss 3325   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    < clt 9414    <_ cle 9415   -ucneg 9592    / cdiv 9989   RR+crp 10987   abscabs 12719    ~~> r crli 12959    ^c ccxp 21950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-xms 19795  df-ms 19796  df-tms 19797  df-cncf 20354  df-limc 21241  df-dv 21242  df-log 21951  df-cxp 21952
This theorem is referenced by:  sqrlim  22309  signsplypnf  26865
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