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

Theorem cxplim 22377
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 11009 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  RR )
21adantl 466 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  x  e.  RR )
3 rpge0 11015 . . . . . 6  |-  ( x  e.  RR+  ->  0  <_  x )
43adantl 466 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  0  <_  x )
5 rpre 11009 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  RR )
65renegcld 9787 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  e.  RR )
76adantr 465 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  e.  RR )
8 rpcn 11011 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  CC )
9 rpne0 11018 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  =/=  0 )
108, 9negne0d 9729 . . . . . . 7  |-  ( A  e.  RR+  ->  -u A  =/=  0 )
1110adantr 465 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  -u A  =/=  0 )
127, 11rereccld 10170 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR+ )  ->  (
1  /  -u A
)  e.  RR )
132, 4, 12recxpcld 22180 . . . 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 22187 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  A )  e.  RR+ )
1716rpreccld 11049 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^c  A ) )  e.  RR+ )
1817rprege0d 11046 . . . . . . . 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 12797 . . . . . . . 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 11026 . . . . . . . . . . . . . 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 11041 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  A )  e.  CC )
2621, 25cxprecd 22186 . . . . . . . . . . 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 11011 . . . . . . . . . . . . 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 11018 . . . . . . . . . . . . 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 22172 . . . . . . . . . . 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 9414 . . . . . . . . . . . . 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 10133 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
3635oveq2d 6119 . . . . . . . . . . 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 2481 . . . . . . . . . 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 10114 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
3938oveq2d 6119 . . . . . . . . . . 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 22191 . . . . . . . . . . 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 11041 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  n  e.  CC )
4241cxp1d 22163 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  1 )  =  n )
4339, 40, 423eqtr3d 2483 . . . . . . . . . 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 4334 . . . . . . . . 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 11026 . . . . . . . . . . . 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 11039 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  x )  e.  RR )
4846rpge0d 11043 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  0  <_  ( 1  /  x
) )
4916rpred 11039 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
n  ^c  A )  e.  RR )
5016rpge0d 11043 . . . . . . . . . 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 22183 . . . . . . . . 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 11059 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( x  ^c 
( 1  /  -u A
) )  <  n
) )  ->  (
1  /  ( n  ^c  A ) )  <  x )
5420, 53eqbrtrd 4324 . . . . . 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 2811 . . . 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 4307 . . . . . . 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 2747 . . . . 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 3085 . . . 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 2811 . 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 22133 . . . . . . 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 11049 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^c  A ) )  e.  RR+ )
6766rpcnd 11041 . . . 4  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
1  /  ( n  ^c  A ) )  e.  CC )
6867ralrimiva 2811 . . 3  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( 1  / 
( n  ^c  A ) )  e.  CC )
69 rpssre 11013 . . . 4  |-  RR+  C_  RR
7069a1i 11 . . 3  |-  ( A  e.  RR+  ->  RR+  C_  RR )
7168, 70rlim0lt 12999 . 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 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   E.wrex 2728    C_ wss 3340   class class class wbr 4304    e. cmpt 4362   ` cfv 5430  (class class class)co 6103   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295    x. cmul 9299    < clt 9430    <_ cle 9431   -ucneg 9608    / cdiv 10005   RR+crp 11003   abscabs 12735    ~~> r crli 12975    ^c ccxp 22019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-fi 7673  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-ioo 11316  df-ioc 11317  df-ico 11318  df-icc 11319  df-fz 11450  df-fzo 11561  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-shft 12568  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-limsup 12961  df-clim 12978  df-rlim 12979  df-sum 13176  df-ef 13365  df-sin 13367  df-cos 13368  df-pi 13370  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-rest 14373  df-topn 14374  df-0g 14392  df-gsum 14393  df-topgen 14394  df-pt 14395  df-prds 14398  df-xrs 14452  df-qtop 14457  df-imas 14458  df-xps 14460  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-submnd 15477  df-mulg 15560  df-cntz 15847  df-cmn 16291  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-fbas 17826  df-fg 17827  df-cnfld 17831  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-cld 18635  df-ntr 18636  df-cls 18637  df-nei 18714  df-lp 18752  df-perf 18753  df-cn 18843  df-cnp 18844  df-haus 18931  df-tx 19147  df-hmeo 19340  df-fil 19431  df-fm 19523  df-flim 19524  df-flf 19525  df-xms 19907  df-ms 19908  df-tms 19909  df-cncf 20466  df-limc 21353  df-dv 21354  df-log 22020  df-cxp 22021
This theorem is referenced by:  sqrlim  22378  signsplypnf  26963
  Copyright terms: Public domain W3C validator