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Theorem metdscn2 18377
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 2296 . . . . . . 7  |-  ( dist `  RR* s )  =  ( dist `  RR* s
)
21xrsdsre 18332 . . . . . 6  |-  ( (
dist `  RR* s )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
31xrsxmet 18331 . . . . . . 7  |-  ( dist `  RR* s )  e.  ( * Met `  RR* )
4 ressxr 8892 . . . . . . 7  |-  RR  C_  RR*
5 eqid 2296 . . . . . . . 8  |-  ( (
dist `  RR* s )  |`  ( RR  X.  RR ) )  =  ( ( dist `  RR* s
)  |`  ( RR  X.  RR ) )
6 eqid 2296 . . . . . . . 8  |-  ( MetOpen `  ( dist `  RR* s ) )  =  ( MetOpen `  ( dist `  RR* s ) )
7 eqid 2296 . . . . . . . 8  |-  ( MetOpen `  ( ( dist `  RR* s
)  |`  ( RR  X.  RR ) ) )  =  ( MetOpen `  ( ( dist `  RR* s )  |`  ( RR  X.  RR ) ) )
85, 6, 7metrest 18086 . . . . . . 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 18318 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( MetOpen `  ( dist ` 
RR* s ) )t  RR )
11 metdscn2.k . . . . . 6  |-  K  =  ( TopOpen ` fld )
1211tgioo2 18325 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( Kt  RR )
1310, 12eqtr3i 2318 . . . 4  |-  ( (
MetOpen `  ( dist `  RR* s
) )t  RR )  =  ( Kt  RR )
1413oveq2i 5885 . . 3  |-  ( J  Cn  ( ( MetOpen `  ( dist `  RR* s ) )t  RR ) )  =  ( J  Cn  ( Kt  RR ) )
1511cnfldtop 18309 . . . 4  |-  K  e. 
Top
16 cnrest2r 17031 . . . 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 3221 . 2  |-  ( J  Cn  ( ( MetOpen `  ( dist `  RR* s ) )t  RR ) )  C_  ( J  Cn  K
)
19 metxmet 17915 . . . . 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 18376 . . . . 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 18373 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X
--> RR )
26 frn 5411 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
276mopntopon 18001 . . . . . 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 17030 . . . . 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 3191 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 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   ran crn 4706    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   RR*cxr 8882    < clt 8883   (,)cioo 10672   distcds 13233   ↾t crest 13341   TopOpenctopn 13342   topGenctg 13358   RR* scxrs 13415   * Metcxmt 16385   Metcme 16386   MetOpencmopn 16388  ℂfldccnfld 16393   Topctop 16647  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  lebnumlem2  18476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xrs 13419  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-xms 17901  df-ms 17902
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