MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cxplim Structured version   Unicode version

Theorem cxplim 23126
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 11227 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  RR )
21adantl 466 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  x  e.  RR )
3 rpge0 11233 . . . . . 6  |-  ( x  e.  RR+  ->  0  <_  x )
43adantl 466 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  0  <_  x )
5 rpre 11227 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  RR )
65renegcld 9987 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  e.  RR )
76adantr 465 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  e.  RR )
8 rpcn 11229 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  CC )
9 rpne0 11236 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
108, 9negne0d 9929 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  =/=  0 )
1110adantr 465 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  =/=  0 )
127, 11rereccld 10372 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
1  /  -u A
)  e.  RR )
132, 4, 12recxpcld 22929 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
x  ^c  ( 1  /  -u A
) )  e.  RR )
14 simprl 755 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  RR+ )
155ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  RR )
1614, 15rpcxpcld 22936 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  A )  e.  RR+ )
1716rpreccld 11267 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^c  A ) )  e.  RR+ )
1817rprege0d 11264 . . . . . . . 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 13095 . . . . . . . 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 754 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  RR+ )
22 simprr 756 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
x  ^c  ( 1  /  -u A
) )  <  n
)
23 rpreccl 11244 . . . . . . . . . . . . . 14  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
2423ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  RR+ )
2524rpcnd 11259 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  CC )
2621, 25cxprecd 22935 . . . . . . . . . . 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 11229 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  CC )
2827ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  x  e.  CC )
29 rpne0 11236 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  =/=  0 )
3029ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  x  =/=  0 )
3128, 30, 25cxpnegd 22921 . . . . . . . . . . 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 9613 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  1  e.  CC )
338ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  A  e.  CC )
349ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  A  =/=  0 )
3532, 33, 34divneg2d 10335 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
3635oveq2d 6301 . . . . . . . . . . 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 2514 . . . . . . . . . 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 10316 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
3938oveq2d 6301 . . . . . . . . . . 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 22940 . . . . . . . . . . 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 11259 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  CC )
4241cxp1d 22912 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  1 )  =  n )
4339, 40, 423eqtr3d 2516 . . . . . . . . . 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 4477 . . . . . . . . 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 11244 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
4645ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR+ )
4746rpred 11257 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR )
4846rpge0d 11261 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( 1  /  x
) )
4916rpred 11257 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  A )  e.  RR )
5016rpge0d 11261 . . . . . . . . . 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 22932 . . . . . . . . 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 11277 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^c  A ) )  <  x )
5420, 53eqbrtrd 4467 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  ( abs `  ( 1  / 
( n  ^c  A ) ) )  <  x )
5554expr 615 . . . . 5  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( x  ^c  ( 1  /  -u A ) )  < 
n  ->  ( abs `  ( 1  /  (
n  ^c  A ) ) )  < 
x ) )
5655ralrimiva 2878 . . . 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 4450 . . . . . . 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 2903 . . . . 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 3214 . . . 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 661 . . 3  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  (
1  /  ( n  ^c  A ) ) )  <  x
) )
6261ralrimiva 2878 . 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 22882 . . . . . . 7  |-  ( ( n  e.  RR+  /\  A  e.  RR )  ->  (
n  ^c  A )  e.  RR+ )
6563, 5, 64syl2anr 478 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^c  A )  e.  RR+ )
6665rpreccld 11267 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^c  A ) )  e.  RR+ )
6766rpcnd 11259 . . . 4  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^c  A ) )  e.  CC )
6867ralrimiva 2878 . . 3  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( 1  / 
( n  ^c  A ) )  e.  CC )
69 rpssre 11231 . . . 4  |-  RR+  C_  RR
7069a1i 11 . . 3  |-  ( A  e.  RR+  ->  RR+  C_  RR )
7168, 70rlim0lt 13298 . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494    x. cmul 9498    < clt 9629    <_ cle 9630   -ucneg 9807    / cdiv 10207   RR+crp 11221   abscabs 13033    ~~> r crli 13274    ^c ccxp 22768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-fi 7872  df-sup 7902  df-oi 7936  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-ioo 11534  df-ioc 11535  df-ico 11536  df-icc 11537  df-fz 11674  df-fzo 11794  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-fac 12323  df-bc 12350  df-hash 12375  df-shft 12866  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-limsup 13260  df-clim 13277  df-rlim 13278  df-sum 13475  df-ef 13668  df-sin 13670  df-cos 13671  df-pi 13673  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-hom 14582  df-cco 14583  df-rest 14681  df-topn 14682  df-0g 14700  df-gsum 14701  df-topgen 14702  df-pt 14703  df-prds 14706  df-xrs 14760  df-qtop 14765  df-imas 14766  df-xps 14768  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-submnd 15790  df-mulg 15874  df-cntz 16169  df-cmn 16615  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-fbas 18227  df-fg 18228  df-cnfld 18232  df-top 19206  df-bases 19208  df-topon 19209  df-topsp 19210  df-cld 19326  df-ntr 19327  df-cls 19328  df-nei 19405  df-lp 19443  df-perf 19444  df-cn 19534  df-cnp 19535  df-haus 19622  df-tx 19890  df-hmeo 20083  df-fil 20174  df-fm 20266  df-flim 20267  df-flf 20268  df-xms 20650  df-ms 20651  df-tms 20652  df-cncf 21209  df-limc 22097  df-dv 22098  df-log 22769  df-cxp 22770
This theorem is referenced by:  sqrtlim  23127  signsplypnf  28258
  Copyright terms: Public domain W3C validator