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Theorem metdscn2 18887
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 2436 . . . . . . 7  |-  ( dist `  RR* s )  =  ( dist `  RR* s
)
21xrsdsre 18841 . . . . . 6  |-  ( (
dist `  RR* s )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
31xrsxmet 18840 . . . . . . 7  |-  ( dist `  RR* s )  e.  ( * Met `  RR* )
4 ressxr 9129 . . . . . . 7  |-  RR  C_  RR*
5 eqid 2436 . . . . . . . 8  |-  ( (
dist `  RR* s )  |`  ( RR  X.  RR ) )  =  ( ( dist `  RR* s
)  |`  ( RR  X.  RR ) )
6 eqid 2436 . . . . . . . 8  |-  ( MetOpen `  ( dist `  RR* s ) )  =  ( MetOpen `  ( dist `  RR* s ) )
7 eqid 2436 . . . . . . . 8  |-  ( MetOpen `  ( ( dist `  RR* s
)  |`  ( RR  X.  RR ) ) )  =  ( MetOpen `  ( ( dist `  RR* s )  |`  ( RR  X.  RR ) ) )
85, 6, 7metrest 18554 . . . . . . 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 654 . . . . . 6  |-  ( (
MetOpen `  ( dist `  RR* s
) )t  RR )  =  (
MetOpen `  ( ( dist `  RR* s )  |`  ( RR  X.  RR ) ) )
102, 9tgioo 18827 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( MetOpen `  ( dist ` 
RR* s ) )t  RR )
11 metdscn2.k . . . . . 6  |-  K  =  ( TopOpen ` fld )
1211tgioo2 18834 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( Kt  RR )
1310, 12eqtr3i 2458 . . . 4  |-  ( (
MetOpen `  ( dist `  RR* s
) )t  RR )  =  ( Kt  RR )
1413oveq2i 6092 . . 3  |-  ( J  Cn  ( ( MetOpen `  ( dist `  RR* s ) )t  RR ) )  =  ( J  Cn  ( Kt  RR ) )
1511cnfldtop 18818 . . . 4  |-  K  e. 
Top
16 cnrest2r 17351 . . . 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 3378 . 2  |-  ( J  Cn  ( ( MetOpen `  ( dist `  RR* s ) )t  RR ) )  C_  ( J  Cn  K
)
19 metxmet 18364 . . . . 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 18886 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR* s
) ) ) )
2319, 22sylan 458 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR* s
) ) ) )
24233adant3 977 . . 3  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR* s
) ) ) )
2520metdsre 18883 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X
--> RR )
26 frn 5597 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
276mopntopon 18469 . . . . . 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 17350 . . . . 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 1270 . . . 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 19 . . 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 202 . 2  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  (
( MetOpen `  ( dist ` 
RR* s ) )t  RR ) ) )
3318, 32sseldi 3346 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 177    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599    C_ wss 3320   (/)c0 3628    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   ran crn 4879    |` cres 4880   -->wf 5450   ` cfv 5454  (class class class)co 6081   supcsup 7445   RRcr 8989   RR*cxr 9119    < clt 9120   (,)cioo 10916   distcds 13538   ↾t crest 13648   TopOpenctopn 13649   topGenctg 13665   RR* scxrs 13722   * Metcxmt 16686   Metcme 16687   MetOpencmopn 16691  ℂfldccnfld 16703   Topctop 16958  TopOnctopon 16959    Cn ccn 17288
This theorem is referenced by:  lebnumlem2  18987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-ec 6907  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-icc 10923  df-fz 11044  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-rest 13650  df-topn 13651  df-topgen 13667  df-xrs 13726  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cn 17291  df-cnp 17292  df-xms 18350  df-ms 18351
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