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Theorem metdscn 18847
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 18839 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
3 iccssxr 10957 . . 3  |-  ( 0 [,]  +oo )  C_  RR*
4 fss 5566 . . 3  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  F : X
--> RR* )
52, 3, 4sylancl 644 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> RR* )
6 simprr 734 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR+ )
75ad2antrr 707 . . . . . . . . 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 737 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  z  e.  X )
97, 8ffvelrnd 5838 . . . . . . . 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* )
10 simprl 733 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  w  e.  X )
117, 10ffvelrnd 5838 . . . . . . . 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* )
12 metdscn.c . . . . . . . . 9  |-  C  =  ( dist `  RR* s
)
1312xrsdsval 16705 . . . . . . . 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 ) ) ) )
149, 11, 13syl2anc 643 . . . . . . 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 ) ) ) )
15 metdscn.j . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
16 metdscn.k . . . . . . . . 9  |-  K  =  ( MetOpen `  C )
17 simplll 735 . . . . . . . . 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
) )
18 simpllr 736 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  S  C_  X
)
19 simplrr 738 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  /\  ( w  e.  X  /\  ( z D w )  <  r ) )  ->  r  e.  RR+ )
20 xmetsym 18338 . . . . . . . . . . 11  |-  ( ( D  e.  ( * Met `  X )  /\  w  e.  X  /\  z  e.  X
)  ->  ( w D z )  =  ( z D w ) )
2117, 10, 8, 20syl3anc 1184 . . . . . . . . . 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 ) )
22 simprr 734 . . . . . . . . . 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 )
2321, 22eqbrtrd 4200 . . . . . . . . 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 )
241, 15, 12, 16, 17, 18, 10, 8, 19, 23metdscnlem 18846 . . . . . . . 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 )
251, 15, 12, 16, 17, 18, 8, 10, 19, 22metdscnlem 18846 . . . . . . . 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 )
26 breq1 4183 . . . . . . . . 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 ) )
27 breq1 4183 . . . . . . . . 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 ) )
2826, 27ifboth 3738 . . . . . . . 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 )
2924, 25, 28syl2anc 643 . . . . . . 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 )
3014, 29eqbrtrd 4200 . . . . . 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 )
3130expr 599 . . . . 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 ) )
3231ralrimiva 2757 . . . 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 ) )
33 breq2 4184 . . . . . . 7  |-  ( s  =  r  ->  (
( z D w )  <  s  <->  ( z D w )  < 
r ) )
3433imbi1d 309 . . . . . 6  |-  ( s  =  r  ->  (
( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r )  <-> 
( ( z D w )  <  r  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) ) )
3534ralbidv 2694 . . . . 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 ) ) )
3635rspcev 3020 . . . 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 ) )
376, 32, 36syl2anc 643 . . 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 ) )
3837ralrimivva 2766 . 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 ) )
39 simpl 444 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  D  e.  ( * Met `  X
) )
4012xrsxmet 18801 . . 3  |-  C  e.  ( * Met `  RR* )
4115, 16metcn 18534 . . 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 ) ) ) )
4239, 40, 41sylancl 644 . 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 ) ) ) )
435, 38, 42mpbir2and 889 1  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675    C_ wss 3288   ifcif 3707   class class class wbr 4180    e. cmpt 4234   `'ccnv 4844   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   supcsup 7411   0cc0 8954    +oocpnf 9081   RR*cxr 9083    < clt 9084    <_ cle 9085   RR+crp 10576    - ecxne 10671   + ecxad 10672   [,]cicc 10883   distcds 13501   RR* scxrs 13685   * Metcxmt 16649   MetOpencmopn 16654    Cn ccn 17250
This theorem is referenced by:  metdscn2  18848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-ec 6874  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-icc 10887  df-fz 11008  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-mulr 13506  df-tset 13511  df-ple 13512  df-ds 13514  df-topgen 13630  df-xrs 13689  df-psmet 16657  df-xmet 16658  df-bl 16660  df-mopn 16661  df-top 16926  df-bases 16928  df-topon 16929  df-cn 17253  df-cnp 17254
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