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Theorem xrge0pluscn 27747
Description: The addition operation of the extended nonnegative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
Hypotheses
Ref Expression
xrge0iifhmeo.1  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 , +oo ,  -u ( log `  x
) ) )
xrge0iifhmeo.k  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
xrge0pluscn.1  |-  .+  =  ( +e  |`  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) )
Assertion
Ref Expression
xrge0pluscn  |-  .+  e.  ( ( J  tX  J )  Cn  J
)
Distinct variable group:    x, F
Allowed substitution hints:    .+ ( x)    J( x)

Proof of Theorem xrge0pluscn
Dummy variables  y  u  v  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrge0iifhmeo.1 . . 3  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 , +oo ,  -u ( log `  x
) ) )
2 xrge0iifhmeo.k . . 3  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
31, 2xrge0iifhmeo 27743 . 2  |-  F  e.  ( II Homeo J )
4 unitsscn 27703 . . . . 5  |-  ( 0 [,] 1 )  C_  CC
5 xpss12 5114 . . . . 5  |-  ( ( ( 0 [,] 1
)  C_  CC  /\  (
0 [,] 1 ) 
C_  CC )  -> 
( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  C_  ( CC  X.  CC ) )
64, 4, 5mp2an 672 . . . 4  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( CC  X.  CC )
7 ax-mulf 9584 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
8 ffn 5737 . . . . 5  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
9 fnssresb 5699 . . . . 5  |-  (  x.  Fn  ( CC  X.  CC )  ->  ( (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( CC  X.  CC ) ) )
107, 8, 9mp2b 10 . . . 4  |-  ( (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( CC  X.  CC ) )
116, 10mpbir 209 . . 3  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  Fn  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) )
12 ovres 6437 . . . . 5  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  =  ( u  x.  v ) )
13 iimulcl 21305 . . . . 5  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u  x.  v )  e.  ( 0 [,] 1 ) )
1412, 13eqeltrd 2555 . . . 4  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  e.  ( 0 [,] 1 ) )
1514rgen2a 2894 . . 3  |-  A. u  e.  ( 0 [,] 1
) A. v  e.  ( 0 [,] 1
) ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  e.  ( 0 [,] 1 )
16 ffnov 6401 . . 3  |-  ( (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> ( 0 [,] 1 )  <->  ( (  x.  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  Fn  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  /\  A. u  e.  ( 0 [,] 1
) A. v  e.  ( 0 [,] 1
) ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  e.  ( 0 [,] 1 ) ) )
1711, 15, 16mpbir2an 918 . 2  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) : ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) --> ( 0 [,] 1
)
18 iccssxr 11619 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
19 xpss12 5114 . . . . . 6  |-  ( ( ( 0 [,] +oo )  C_  RR*  /\  (
0 [,] +oo )  C_ 
RR* )  ->  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) )  C_  ( RR*  X.  RR* ) )
2018, 18, 19mp2an 672 . . . . 5  |-  ( ( 0 [,] +oo )  X.  ( 0 [,] +oo ) )  C_  ( RR*  X.  RR* )
21 xaddf 11435 . . . . . 6  |-  +e : ( RR*  X.  RR* )
--> RR*
22 ffn 5737 . . . . . 6  |-  ( +e : ( RR*  X. 
RR* ) --> RR*  ->  +e  Fn  ( RR*  X. 
RR* ) )
23 fnssresb 5699 . . . . . 6  |-  ( +e  Fn  ( RR*  X. 
RR* )  ->  (
( +e  |`  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) )  Fn  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) )  <->  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) )  C_  ( RR*  X.  RR* ) ) )
2421, 22, 23mp2b 10 . . . . 5  |-  ( ( +e  |`  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) )  Fn  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
)  <->  ( ( 0 [,] +oo )  X.  ( 0 [,] +oo ) )  C_  ( RR*  X.  RR* ) )
2520, 24mpbir 209 . . . 4  |-  ( +e  |`  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) ) )  Fn  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
)
26 xrge0pluscn.1 . . . . 5  |-  .+  =  ( +e  |`  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) )
2726fneq1i 5681 . . . 4  |-  (  .+  Fn  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
)  <->  ( +e  |`  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) )  Fn  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) )
2825, 27mpbir 209 . . 3  |-  .+  Fn  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
)
2926oveqi 6308 . . . . 5  |-  ( a 
.+  b )  =  ( a ( +e  |`  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) b )
30 ovres 6437 . . . . . 6  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a ( +e  |`  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) b )  =  ( a +e b ) )
31 ge0xaddcl 11646 . . . . . 6  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a +e b )  e.  ( 0 [,] +oo ) )
3230, 31eqeltrd 2555 . . . . 5  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a ( +e  |`  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) b )  e.  ( 0 [,] +oo ) )
3329, 32syl5eqel 2559 . . . 4  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a  .+  b )  e.  ( 0 [,] +oo )
)
3433rgen2a 2894 . . 3  |-  A. a  e.  ( 0 [,] +oo ) A. b  e.  ( 0 [,] +oo )
( a  .+  b
)  e.  ( 0 [,] +oo )
35 ffnov 6401 . . 3  |-  (  .+  : ( ( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) --> ( 0 [,] +oo )  <->  (  .+  Fn  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
)  /\  A. a  e.  ( 0 [,] +oo ) A. b  e.  ( 0 [,] +oo )
( a  .+  b
)  e.  ( 0 [,] +oo ) ) )
3628, 34, 35mpbir2an 918 . 2  |-  .+  :
( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) --> ( 0 [,] +oo )
37 iitopon 21251 . 2  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
38 letopon 19574 . . . 4  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
39 resttopon 19530 . . . 4  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ( 0 [,] +oo )  C_  RR* )  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  (
0 [,] +oo )
) )
4038, 18, 39mp2an 672 . . 3  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  ( 0 [,] +oo ) )
412, 40eqeltri 2551 . 2  |-  J  e.  (TopOn `  ( 0 [,] +oo ) )
4226oveqi 6308 . . . 4  |-  ( ( F `  u ) 
.+  ( F `  v ) )  =  ( ( F `  u ) ( +e  |`  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) ( F `  v ) )
431xrge0iifcnv 27740 . . . . . . . 8  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  /\  `' F  =  ( y  e.  ( 0 [,] +oo )  |->  if ( y  = +oo ,  0 ,  ( exp `  -u y
) ) ) )
4443simpli 458 . . . . . . 7  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,] +oo )
45 f1of 5822 . . . . . . 7  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo )  ->  F : ( 0 [,] 1 ) --> ( 0 [,] +oo ) )
4644, 45ax-mp 5 . . . . . 6  |-  F :
( 0 [,] 1
) --> ( 0 [,] +oo )
4746ffvelrni 6031 . . . . 5  |-  ( u  e.  ( 0 [,] 1 )  ->  ( F `  u )  e.  ( 0 [,] +oo ) )
4846ffvelrni 6031 . . . . 5  |-  ( v  e.  ( 0 [,] 1 )  ->  ( F `  v )  e.  ( 0 [,] +oo ) )
49 ovres 6437 . . . . 5  |-  ( ( ( F `  u
)  e.  ( 0 [,] +oo )  /\  ( F `  v )  e.  ( 0 [,] +oo ) )  ->  (
( F `  u
) ( +e  |`  ( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) ) ( F `
 v ) )  =  ( ( F `
 u ) +e ( F `  v ) ) )
5047, 48, 49syl2an 477 . . . 4  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( ( F `
 u ) ( +e  |`  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) ( F `  v ) )  =  ( ( F `  u ) +e ( F `
 v ) ) )
5142, 50syl5eq 2520 . . 3  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( ( F `
 u )  .+  ( F `  v ) )  =  ( ( F `  u ) +e ( F `
 v ) ) )
521, 2xrge0iifhom 27744 . . 3  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( F `  ( u  x.  v
) )  =  ( ( F `  u
) +e ( F `  v ) ) )
5312eqcomd 2475 . . . 4  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u  x.  v )  =  ( u (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) ) v ) )
5453fveq2d 5876 . . 3  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( F `  ( u  x.  v
) )  =  ( F `  ( u (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) ) v ) ) )
5551, 52, 543eqtr2rd 2515 . 2  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( F `  ( u (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) v ) )  =  ( ( F `  u ) 
.+  ( F `  v ) ) )
56 eqid 2467 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  ( (mulGrp ` fld )s  ( 0 [,] 1 ) )
5756iistmd 27709 . . 3  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  e. TopMnd
58 cnfldex 18293 . . . . . 6  |-fld  e.  _V
59 ovex 6320 . . . . . 6  |-  ( 0 [,] 1 )  e. 
_V
60 eqid 2467 . . . . . . 7  |-  (flds  ( 0 [,] 1 ) )  =  (flds  ( 0 [,] 1 ) )
61 eqid 2467 . . . . . . 7  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
6260, 61mgpress 17024 . . . . . 6  |-  ( (fld  e. 
_V  /\  ( 0 [,] 1 )  e. 
_V )  ->  (
(mulGrp ` fld )s  ( 0 [,] 1 ) )  =  (mulGrp `  (flds  ( 0 [,] 1 ) ) ) )
6358, 59, 62mp2an 672 . . . . 5  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  (mulGrp `  (flds  ( 0 [,] 1 ) ) )
6460dfii4 21256 . . . . 5  |-  II  =  ( TopOpen `  (flds  ( 0 [,] 1 ) ) )
6563, 64mgptopn 17022 . . . 4  |-  II  =  ( TopOpen `  ( (mulGrp ` fld )s  ( 0 [,] 1 ) ) )
66 cnfldbas 18294 . . . . . . 7  |-  CC  =  ( Base ` fld )
6761, 66mgpbas 17019 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
68 cnfldmul 18296 . . . . . . 7  |-  x.  =  ( .r ` fld )
6961, 68mgpplusg 17017 . . . . . 6  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
707, 8ax-mp 5 . . . . . 6  |-  x.  Fn  ( CC  X.  CC )
7167, 56, 69, 70, 4ressplusf 27462 . . . . 5  |-  ( +f `  ( (mulGrp ` fld )s  ( 0 [,] 1
) ) )  =  (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
7271eqcomi 2480 . . . 4  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  =  ( +f `  (
(mulGrp ` fld )s  ( 0 [,] 1 ) ) )
7365, 72tmdcn 20450 . . 3  |-  ( ( (mulGrp ` fld )s  ( 0 [,] 1 ) )  e. TopMnd  ->  (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  e.  ( ( II  tX  II )  Cn  II ) )
7457, 73ax-mp 5 . 2  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  ( ( II  tX  II )  Cn  II )
753, 17, 36, 37, 41, 55, 74mndpluscn 27733 1  |-  .+  e.  ( ( J  tX  J )  Cn  J
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    C_ wss 3481   ifcif 3945    |-> cmpt 4511    X. cxp 5003   `'ccnv 5004    |` cres 5007    Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    x. cmul 9509   +oocpnf 9637   RR*cxr 9639    <_ cle 9641   -ucneg 9818   +ecxad 11328   [,]cicc 11544   expce 13676   ↾s cress 14508   ↾t crest 14693  ordTopcordt 14771   +fcplusf 15743  mulGrpcmgp 17013  ℂfldccnfld 18290  TopOnctopon 19264    Cn ccn 19593    tX ctx 19929  TopMndctmd 20437   IIcii 21247   logclog 22808
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-pi 13687  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-ordt 14773  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-ps 15704  df-tsr 15705  df-plusf 15745  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-subrg 17298  df-abv 17337  df-lmod 17385  df-scaf 17386  df-sra 17689  df-rgmod 17690  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-tmd 20439  df-tgp 20440  df-trg 20530  df-xms 20691  df-ms 20692  df-tms 20693  df-nm 20971  df-ngp 20972  df-nrg 20974  df-nlm 20975  df-ii 21249  df-cncf 21250  df-limc 22138  df-dv 22139  df-log 22810
This theorem is referenced by:  xrge0tmdOLD  27752
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