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Theorem metdscn2 21530
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 complex numbers. (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 2454 . . . . . . 7  |-  ( dist `  RR*s )  =  ( dist `  RR*s
)
21xrsdsre 21484 . . . . . 6  |-  ( (
dist `  RR*s )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
31xrsxmet 21483 . . . . . . 7  |-  ( dist `  RR*s )  e.  ( *Met `  RR* )
4 ressxr 9626 . . . . . . 7  |-  RR  C_  RR*
5 eqid 2454 . . . . . . . 8  |-  ( (
dist `  RR*s )  |`  ( RR  X.  RR ) )  =  ( ( dist `  RR*s
)  |`  ( RR  X.  RR ) )
6 eqid 2454 . . . . . . . 8  |-  ( MetOpen `  ( dist `  RR*s ) )  =  ( MetOpen `  ( dist `  RR*s ) )
7 eqid 2454 . . . . . . . 8  |-  ( MetOpen `  ( ( dist `  RR*s
)  |`  ( RR  X.  RR ) ) )  =  ( MetOpen `  ( ( dist `  RR*s )  |`  ( RR  X.  RR ) ) )
85, 6, 7metrest 21196 . . . . . . 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 670 . . . . . 6  |-  ( (
MetOpen `  ( dist `  RR*s
) )t  RR )  =  (
MetOpen `  ( ( dist `  RR*s )  |`  ( RR  X.  RR ) ) )
102, 9tgioo 21470 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( MetOpen `  ( dist ` 
RR*s ) )t  RR )
11 metdscn2.k . . . . . 6  |-  K  =  ( TopOpen ` fld )
1211tgioo2 21477 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( Kt  RR )
1310, 12eqtr3i 2485 . . . 4  |-  ( (
MetOpen `  ( dist `  RR*s
) )t  RR )  =  ( Kt  RR )
1413oveq2i 6281 . . 3  |-  ( J  Cn  ( ( MetOpen `  ( dist `  RR*s ) )t  RR ) )  =  ( J  Cn  ( Kt  RR ) )
1511cnfldtop 21460 . . . 4  |-  K  e. 
Top
16 cnrest2r 19958 . . . 4  |-  ( K  e.  Top  ->  ( J  Cn  ( Kt  RR ) )  C_  ( J  Cn  K ) )
1715, 16ax-mp 5 . . 3  |-  ( J  Cn  ( Kt  RR ) )  C_  ( J  Cn  K )
1814, 17eqsstri 3519 . 2  |-  ( J  Cn  ( ( MetOpen `  ( dist `  RR*s ) )t  RR ) )  C_  ( J  Cn  K
)
19 metxmet 21006 . . . . 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 21529 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR*s
) ) ) )
2319, 22sylan 469 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR*s
) ) ) )
24233adant3 1014 . . 3  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  ( MetOpen
`  ( dist `  RR*s
) ) ) )
2520metdsre 21526 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X
--> RR )
26 frn 5719 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
276mopntopon 21111 . . . . . 6  |-  ( (
dist `  RR*s )  e.  ( *Met ` 
RR* )  ->  ( MetOpen
`  ( dist `  RR*s
) )  e.  (TopOn `  RR* ) )
283, 27ax-mp 5 . . . . 5  |-  ( MetOpen `  ( dist `  RR*s ) )  e.  (TopOn `  RR* )
29 cnrest2 19957 . . . . 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 1313 . . . 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 20 . . 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 210 . 2  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  (
( MetOpen `  ( dist ` 
RR*s ) )t  RR ) ) )
3318, 32sseldi 3487 1  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649    C_ wss 3461   (/)c0 3783    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   ran crn 4989    |` cres 4990   -->wf 5566   ` cfv 5570  (class class class)co 6270   supcsup 7892   RRcr 9480   RR*cxr 9616    < clt 9617   (,)cioo 11532   distcds 14796   ↾t crest 14913   TopOpenctopn 14914   topGenctg 14930   RR*scxrs 14992   *Metcxmt 18601   Metcme 18602   MetOpencmopn 18606  ℂfldccnfld 18618   Topctop 19564  TopOnctopon 19565    Cn ccn 19895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-ec 7305  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-icc 11539  df-fz 11676  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-plusg 14800  df-mulr 14801  df-starv 14802  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-rest 14915  df-topn 14916  df-topgen 14936  df-xrs 14994  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cn 19898  df-cnp 19899  df-xms 20992  df-ms 20993
This theorem is referenced by:  lebnumlem2  21631
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