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Theorem xrge0pluscn 26385
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 26381 . 2  |-  F  e.  ( II Homeo J )
4 unitsscn 26341 . . . . 5  |-  ( 0 [,] 1 )  C_  CC
5 xpss12 4960 . . . . 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 9377 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
8 ffn 5574 . . . . 5  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
9 fnssresb 5538 . . . . 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 6245 . . . . 5  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  =  ( u  x.  v ) )
13 iimulcl 20524 . . . . 5  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u  x.  v )  e.  ( 0 [,] 1 ) )
1412, 13eqeltrd 2517 . . . 4  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  e.  ( 0 [,] 1 ) )
1514rgen2a 2797 . . 3  |-  A. u  e.  ( 0 [,] 1
) A. v  e.  ( 0 [,] 1
) ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  e.  ( 0 [,] 1 )
16 ffnov 6209 . . 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 911 . 2  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) : ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) --> ( 0 [,] 1
)
18 iccssxr 11393 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
19 xpss12 4960 . . . . . 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 11209 . . . . . 6  |-  +e : ( RR*  X.  RR* )
--> RR*
22 ffn 5574 . . . . . 6  |-  ( +e : ( RR*  X. 
RR* ) --> RR*  ->  +e  Fn  ( RR*  X. 
RR* ) )
23 fnssresb 5538 . . . . . 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 5520 . . . 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 6119 . . . . 5  |-  ( a 
.+  b )  =  ( a ( +e  |`  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) b )
30 ovres 6245 . . . . . 6  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a ( +e  |`  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) b )  =  ( a +e b ) )
31 ge0xaddcl 11414 . . . . . 6  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a +e b )  e.  ( 0 [,] +oo ) )
3230, 31eqeltrd 2517 . . . . 5  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a ( +e  |`  (
( 0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) b )  e.  ( 0 [,] +oo ) )
3329, 32syl5eqel 2527 . . . 4  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a  .+  b )  e.  ( 0 [,] +oo )
)
3433rgen2a 2797 . . 3  |-  A. a  e.  ( 0 [,] +oo ) A. b  e.  ( 0 [,] +oo )
( a  .+  b
)  e.  ( 0 [,] +oo )
35 ffnov 6209 . . 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 911 . 2  |-  .+  :
( ( 0 [,] +oo )  X.  (
0 [,] +oo )
) --> ( 0 [,] +oo )
37 iitopon 20470 . 2  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
38 letopon 18824 . . . 4  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
39 resttopon 18780 . . . 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 2513 . 2  |-  J  e.  (TopOn `  ( 0 [,] +oo ) )
4226oveqi 6119 . . . 4  |-  ( ( F `  u ) 
.+  ( F `  v ) )  =  ( ( F `  u ) ( +e  |`  ( (
0 [,] +oo )  X.  ( 0 [,] +oo ) ) ) ( F `  v ) )
431xrge0iifcnv 26378 . . . . . . . 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 5656 . . . . . . 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 5857 . . . . 5  |-  ( u  e.  ( 0 [,] 1 )  ->  ( F `  u )  e.  ( 0 [,] +oo ) )
4846ffvelrni 5857 . . . . 5  |-  ( v  e.  ( 0 [,] 1 )  ->  ( F `  v )  e.  ( 0 [,] +oo ) )
49 ovres 6245 . . . . 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 2487 . . 3  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( ( F `
 u )  .+  ( F `  v ) )  =  ( ( F `  u ) +e ( F `
 v ) ) )
521, 2xrge0iifhom 26382 . . 3  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( F `  ( u  x.  v
) )  =  ( ( F `  u
) +e ( F `  v ) ) )
5312eqcomd 2448 . . . 4  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u  x.  v )  =  ( u (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) ) v ) )
5453fveq2d 5710 . . 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 2482 . 2  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( F `  ( u (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) v ) )  =  ( ( F `  u ) 
.+  ( F `  v ) ) )
56 eqid 2443 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  ( (mulGrp ` fld )s  ( 0 [,] 1 ) )
5756iistmd 26347 . . 3  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  e. TopMnd
58 cnfldex 17836 . . . . . 6  |-fld  e.  _V
59 ovex 6131 . . . . . 6  |-  ( 0 [,] 1 )  e. 
_V
60 eqid 2443 . . . . . . 7  |-  (flds  ( 0 [,] 1 ) )  =  (flds  ( 0 [,] 1 ) )
61 eqid 2443 . . . . . . 7  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
6260, 61mgpress 16617 . . . . . 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 20475 . . . . 5  |-  II  =  ( TopOpen `  (flds  ( 0 [,] 1 ) ) )
6563, 64mgptopn 16615 . . . 4  |-  II  =  ( TopOpen `  ( (mulGrp ` fld )s  ( 0 [,] 1 ) ) )
66 cnfldbas 17837 . . . . . . 7  |-  CC  =  ( Base ` fld )
6761, 66mgpbas 16612 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
68 cnfldmul 17839 . . . . . . 7  |-  x.  =  ( .r ` fld )
6961, 68mgpplusg 16610 . . . . . 6  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
707, 8ax-mp 5 . . . . . 6  |-  x.  Fn  ( CC  X.  CC )
7167, 56, 69, 70, 4ressplusf 26126 . . . . 5  |-  ( +f `  ( (mulGrp ` fld )s  ( 0 [,] 1
) ) )  =  (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
7271eqcomi 2447 . . . 4  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  =  ( +f `  (
(mulGrp ` fld )s  ( 0 [,] 1 ) ) )
7365, 72tmdcn 19669 . . 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 26371 1  |-  .+  e.  ( ( J  tX  J )  Cn  J
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2730   _Vcvv 2987    C_ wss 3343   ifcif 3806    e. cmpt 4365    X. cxp 4853   `'ccnv 4854    |` cres 4857    Fn wfn 5428   -->wf 5429   -1-1-onto->wf1o 5432   ` cfv 5433  (class class class)co 6106   CCcc 9295   0cc0 9297   1c1 9298    x. cmul 9302   +oocpnf 9430   RR*cxr 9432    <_ cle 9434   -ucneg 9611   +ecxad 11102   [,]cicc 11318   expce 13362   ↾s cress 14190   ↾t crest 14374  ordTopcordt 14452   +fcplusf 15427  mulGrpcmgp 16606  ℂfldccnfld 17833  TopOnctopon 18514    Cn ccn 18843    tX ctx 19148  TopMndctmd 19656   IIcii 20466   logclog 22021
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 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376  ax-mulf 9377
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 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-supp 6706  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-map 7231  df-pm 7232  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fsupp 7636  df-fi 7676  df-sup 7706  df-oi 7739  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-q 10969  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-ioo 11319  df-ioc 11320  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-fl 11657  df-mod 11724  df-seq 11822  df-exp 11881  df-fac 12067  df-bc 12094  df-hash 12119  df-shft 12571  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-limsup 12964  df-clim 12981  df-rlim 12982  df-sum 13179  df-ef 13368  df-sin 13370  df-cos 13371  df-pi 13373  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-sca 14269  df-vsca 14270  df-ip 14271  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-hom 14277  df-cco 14278  df-rest 14376  df-topn 14377  df-0g 14395  df-gsum 14396  df-topgen 14397  df-pt 14398  df-prds 14401  df-ordt 14454  df-xrs 14455  df-qtop 14460  df-imas 14461  df-xps 14463  df-mre 14539  df-mrc 14540  df-acs 14542  df-ps 15385  df-tsr 15386  df-mnd 15430  df-plusf 15431  df-submnd 15480  df-grp 15560  df-minusg 15561  df-sbg 15562  df-mulg 15563  df-subg 15693  df-cntz 15850  df-cmn 16294  df-abl 16295  df-mgp 16607  df-ur 16619  df-rng 16662  df-cring 16663  df-subrg 16878  df-abv 16917  df-lmod 16965  df-scaf 16966  df-sra 17268  df-rgmod 17269  df-psmet 17824  df-xmet 17825  df-met 17826  df-bl 17827  df-mopn 17828  df-fbas 17829  df-fg 17830  df-cnfld 17834  df-top 18518  df-bases 18520  df-topon 18521  df-topsp 18522  df-cld 18638  df-ntr 18639  df-cls 18640  df-nei 18717  df-lp 18755  df-perf 18756  df-cn 18846  df-cnp 18847  df-haus 18934  df-tx 19150  df-hmeo 19343  df-fil 19434  df-fm 19526  df-flim 19527  df-flf 19528  df-tmd 19658  df-tgp 19659  df-trg 19749  df-xms 19910  df-ms 19911  df-tms 19912  df-nm 20190  df-ngp 20191  df-nrg 20193  df-nlm 20194  df-ii 20468  df-cncf 20469  df-limc 21356  df-dv 21357  df-log 22023
This theorem is referenced by:  xrge0tmdOLD  26390
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