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Theorem metdscn 18360
Description: The function  F which gives the distance from a point to a set is a continuous function into the metric topology of the extended reals. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-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 )
metdscn.c  |-  C  =  ( dist `  RR* s
)
metdscn.k  |-  K  =  ( MetOpen `  C )
Assertion
Ref Expression
metdscn  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  K
) )
Distinct variable groups:    x, y, D    y, J    x, S, y    x, X, y
Allowed substitution hints:    C( x, y)    F( x, y)    J( x)    K( x, y)

Proof of Theorem metdscn
Dummy variables  w  r  z  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metdscn.f . . . 4  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
21metdsf 18352 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
3 iccssxr 10732 . . 3  |-  ( 0 [,]  +oo )  C_  RR*
4 fss 5397 . . 3  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  F : X
--> RR* )
52, 3, 4sylancl 643 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> RR* )
6 simprr 733 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR+ )
75ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  F : X
--> RR* )
8 simplrl 736 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  z  e.  X )
9 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : X --> RR*  /\  z  e.  X )  ->  ( F `  z )  e.  RR* )
107, 8, 9syl2anc 642 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( F `  z )  e.  RR* )
11 simprl 732 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  w  e.  X )
12 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : X --> RR*  /\  w  e.  X )  ->  ( F `  w )  e.  RR* )
137, 11, 12syl2anc 642 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( F `  w )  e.  RR* )
14 metdscn.c . . . . . . . . 9  |-  C  =  ( dist `  RR* s
)
1514xrsdsval 16415 . . . . . . . 8  |-  ( ( ( F `  z
)  e.  RR*  /\  ( F `  w )  e.  RR* )  ->  (
( F `  z
) C ( F `
 w ) )  =  if ( ( F `  z )  <_  ( F `  w ) ,  ( ( F `  w
) + e  - e ( F `  z ) ) ,  ( ( F `  z ) + e  - e ( F `  w ) ) ) )
1610, 13, 15syl2anc 642 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( ( F `  z ) C ( F `  w ) )  =  if ( ( F `
 z )  <_ 
( F `  w
) ,  ( ( F `  w ) + e  - e
( F `  z
) ) ,  ( ( F `  z
) + e  - e ( F `  w ) ) ) )
17 metdscn.j . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
18 metdscn.k . . . . . . . . 9  |-  K  =  ( MetOpen `  C )
19 simplll 734 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  D  e.  ( * Met `  X
) )
20 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  S  C_  X
)
21 simplrr 737 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  r  e.  RR+ )
22 xmetsym 17912 . . . . . . . . . . 11  |-  ( ( D  e.  ( * Met `  X )  /\  w  e.  X  /\  z  e.  X
)  ->  ( w D z )  =  ( z D w ) )
2319, 11, 8, 22syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( w D z )  =  ( z D w ) )
24 simprr 733 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( z D w )  < 
r )
2523, 24eqbrtrd 4043 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( w D z )  < 
r )
261, 17, 14, 18, 19, 20, 11, 8, 21, 25metdscnlem 18359 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( ( F `  w ) + e  - e ( F `  z ) )  <  r )
271, 17, 14, 18, 19, 20, 8, 11, 21, 24metdscnlem 18359 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( ( F `  z ) + e  - e ( F `  w ) )  <  r )
28 breq1 4026 . . . . . . . . 9  |-  ( ( ( F `  w
) + e  - e ( F `  z ) )  =  if ( ( F `
 z )  <_ 
( F `  w
) ,  ( ( F `  w ) + e  - e
( F `  z
) ) ,  ( ( F `  z
) + e  - e ( F `  w ) ) )  ->  ( ( ( F `  w ) + e  - e
( F `  z
) )  <  r  <->  if ( ( F `  z )  <_  ( F `  w ) ,  ( ( F `
 w ) + e  - e ( F `  z ) ) ,  ( ( F `  z ) + e  - e
( F `  w
) ) )  < 
r ) )
29 breq1 4026 . . . . . . . . 9  |-  ( ( ( F `  z
) + e  - e ( F `  w ) )  =  if ( ( F `
 z )  <_ 
( F `  w
) ,  ( ( F `  w ) + e  - e
( F `  z
) ) ,  ( ( F `  z
) + e  - e ( F `  w ) ) )  ->  ( ( ( F `  z ) + e  - e
( F `  w
) )  <  r  <->  if ( ( F `  z )  <_  ( F `  w ) ,  ( ( F `
 w ) + e  - e ( F `  z ) ) ,  ( ( F `  z ) + e  - e
( F `  w
) ) )  < 
r ) )
3028, 29ifboth 3596 . . . . . . . 8  |-  ( ( ( ( F `  w ) + e  - e ( F `  z ) )  < 
r  /\  ( ( F `  z ) + e  - e ( F `  w ) )  <  r )  ->  if ( ( F `  z )  <_  ( F `  w ) ,  ( ( F `  w
) + e  - e ( F `  z ) ) ,  ( ( F `  z ) + e  - e ( F `  w ) ) )  <  r )
3126, 27, 30syl2anc 642 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  if (
( F `  z
)  <_  ( F `  w ) ,  ( ( F `  w
) + e  - e ( F `  z ) ) ,  ( ( F `  z ) + e  - e ( F `  w ) ) )  <  r )
3216, 31eqbrtrd 4043 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  ( ( F `  z ) C ( F `  w ) )  < 
r )
3332expr 598 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  w  e.  X )  ->  ( ( z D w )  <  r  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) )
3433ralrimiva 2626 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  A. w  e.  X  ( ( z D w )  <  r  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) )
35 breq2 4027 . . . . . . 7  |-  ( s  =  r  ->  (
( z D w )  <  s  <->  ( z D w )  < 
r ) )
3635imbi1d 308 . . . . . 6  |-  ( s  =  r  ->  (
( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r )  <-> 
( ( z D w )  <  r  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) ) )
3736ralbidv 2563 . . . . 5  |-  ( s  =  r  ->  ( A. w  e.  X  ( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r )  <->  A. w  e.  X  ( ( z D w )  <  r  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) ) )
3837rspcev 2884 . . . 4  |-  ( ( r  e.  RR+  /\  A. w  e.  X  (
( z D w )  <  r  -> 
( ( F `  z ) C ( F `  w ) )  <  r ) )  ->  E. s  e.  RR+  A. w  e.  X  ( ( z D w )  < 
s  ->  ( ( F `  z ) C ( F `  w ) )  < 
r ) )
396, 34, 38syl2anc 642 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  E. s  e.  RR+  A. w  e.  X  ( (
z D w )  <  s  ->  (
( F `  z
) C ( F `
 w ) )  <  r ) )
4039ralrimivva 2635 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  A. z  e.  X  A. r  e.  RR+  E. s  e.  RR+  A. w  e.  X  ( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) )
41 simpl 443 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  D  e.  ( * Met `  X
) )
4214xrsxmet 18315 . . 3  |-  C  e.  ( * Met `  RR* )
4317, 18metcn 18089 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  C  e.  ( * Met `  RR* )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> RR*  /\  A. z  e.  X  A. r  e.  RR+  E. s  e.  RR+  A. w  e.  X  ( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) ) ) )
4441, 42, 43sylancl 643 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> RR*  /\  A. z  e.  X  A. r  e.  RR+  E. s  e.  RR+  A. w  e.  X  ( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) ) ) )
455, 40, 44mpbir2and 888 1  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   0cc0 8737    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   RR+crp 10354    - ecxne 10449   + ecxad 10450   [,]cicc 10659   distcds 13217   RR* scxrs 13399   * Metcxmt 16369   MetOpencmopn 16372    Cn ccn 16954
This theorem is referenced by:  metdscn2  18361
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-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-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-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-tset 13227  df-ple 13228  df-ds 13230  df-topgen 13344  df-xrs 13403  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-cnp 16958
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