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Theorem metdscn2 18361
Description: The function  F which gives the distance from a point to a nonempty set in a metric space is a continuous function into the topology of the complexes. (Contributed by Mario Carneiro, 5-Sep-2015.)
Hypotheses
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
metdscn.j  |-  J  =  ( MetOpen `  D )
metdscn2.k  |-  K  =  ( TopOpen ` fld )
Assertion
Ref Expression
metdscn2  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  K
) )
Distinct variable groups:    x, y, D    y, J    x, S, y    x, X, y
Allowed substitution hints:    F( x, y)    J( x)    K( x, y)

Proof of Theorem metdscn2
StepHypRef Expression
1 eqid 2283 . . . . . . 7  |-  ( dist `  RR* s )  =  ( dist `  RR* s
)
21xrsdsre 18316 . . . . . 6  |-  ( (
dist `  RR* s )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
31xrsxmet 18315 . . . . . . 7  |-  ( dist `  RR* s )  e.  ( * Met `  RR* )
4 ressxr 8876 . . . . . . 7  |-  RR  C_  RR*
5 eqid 2283 . . . . . . . 8  |-  ( (
dist `  RR* s )  |`  ( RR  X.  RR ) )  =  ( ( dist `  RR* s
)  |`  ( RR  X.  RR ) )
6 eqid 2283 . . . . . . . 8  |-  ( MetOpen `  ( dist `  RR* s ) )  =  ( MetOpen `  ( dist `  RR* s ) )
7 eqid 2283 . . . . . . . 8  |-  ( MetOpen `  ( ( dist `  RR* s
)  |`  ( RR  X.  RR ) ) )  =  ( MetOpen `  ( ( dist `  RR* s )  |`  ( RR  X.  RR ) ) )
85, 6, 7metrest 18070 . . . . . . 7  |-  ( ( ( dist `  RR* s
)  e.  ( * Met `  RR* )  /\  RR  C_  RR* )  -> 
( ( MetOpen `  ( dist `  RR* s ) )t  RR )  =  ( MetOpen `  ( ( dist `  RR* s
)  |`  ( RR  X.  RR ) ) ) )
93, 4, 8mp2an 653 . . . . . 6  |-  ( (
MetOpen `  ( dist `  RR* s
) )t  RR )  =  (
MetOpen `  ( ( dist `  RR* s )  |`  ( RR  X.  RR ) ) )
102, 9tgioo 18302 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( MetOpen `  ( dist ` 
RR* s ) )t  RR )
11 metdscn2.k . . . . . 6  |-  K  =  ( TopOpen ` fld )
1211tgioo2 18309 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( Kt  RR )
1310, 12eqtr3i 2305 . . . 4  |-  ( (
MetOpen `  ( dist `  RR* s
) )t  RR )  =  ( Kt  RR )
1413oveq2i 5869 . . 3  |-  ( J  Cn  ( ( MetOpen `  ( dist `  RR* s ) )t  RR ) )  =  ( J  Cn  ( Kt  RR ) )
1511cnfldtop 18293 . . . 4  |-  K  e. 
Top
16 cnrest2r 17015 . . . 4  |-  ( K  e.  Top  ->  ( J  Cn  ( Kt  RR ) )  C_  ( J  Cn  K ) )
1715, 16ax-mp 8 . . 3  |-  ( J  Cn  ( Kt  RR ) )  C_  ( J  Cn  K )
1814, 17eqsstri 3208 . 2  |-  ( J  Cn  ( ( MetOpen `  ( dist `  RR* s ) )t  RR ) )  C_  ( J  Cn  K
)
19 metxmet 17899 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
20 metdscn.f . . . . . 6  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
21 metdscn.j . . . . . 6  |-  J  =  ( MetOpen `  D )
2220, 21, 1, 6metdscn 18360 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR* s
) ) ) )
2319, 22sylan 457 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR* s
) ) ) )
24233adant3 975 . . 3  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR* s
) ) ) )
2520metdsre 18357 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X
--> RR )
26 frn 5395 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
276mopntopon 17985 . . . . . 6  |-  ( (
dist `  RR* s )  e.  ( * Met ` 
RR* )  ->  ( MetOpen
`  ( dist `  RR* s
) )  e.  (TopOn `  RR* ) )
283, 27ax-mp 8 . . . . 5  |-  ( MetOpen `  ( dist `  RR* s ) )  e.  (TopOn `  RR* )
29 cnrest2 17014 . . . . 5  |-  ( ( ( MetOpen `  ( dist ` 
RR* s ) )  e.  (TopOn `  RR* )  /\  ran  F  C_  RR  /\  RR  C_  RR* )  -> 
( F  e.  ( J  Cn  ( MetOpen `  ( dist `  RR* s ) ) )  <->  F  e.  ( J  Cn  (
( MetOpen `  ( dist ` 
RR* s ) )t  RR ) ) ) )
3028, 4, 29mp3an13 1268 . . . 4  |-  ( ran 
F  C_  RR  ->  ( F  e.  ( J  Cn  ( MetOpen `  ( dist `  RR* s ) ) )  <->  F  e.  ( J  Cn  ( ( MetOpen `  ( dist `  RR* s ) )t  RR ) ) ) )
3125, 26, 303syl 18 . . 3  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( F  e.  ( J  Cn  ( MetOpen `  ( dist ` 
RR* s ) ) )  <->  F  e.  ( J  Cn  ( ( MetOpen `  ( dist `  RR* s ) )t  RR ) ) ) )
3224, 31mpbid 201 . 2  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  (
( MetOpen `  ( dist ` 
RR* s ) )t  RR ) ) )
3318, 32sseldi 3178 1  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   RR*cxr 8866    < clt 8867   (,)cioo 10656   distcds 13217   ↾t crest 13325   TopOpenctopn 13326   topGenctg 13342   RR* scxrs 13399   * Metcxmt 16369   Metcme 16370   MetOpencmopn 16372  ℂfldccnfld 16377   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  lebnumlem2  18460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xrs 13403  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-xms 17885  df-ms 17886
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