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Theorem metdscn 18917
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 18909 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
3 iccssxr 11024 . . 3  |-  ( 0 [,]  +oo )  C_  RR*
4 fss 5628 . . 3  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  F : X
--> RR* )
52, 3, 4sylancl 645 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> RR* )
6 simprr 735 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR+ )
75ad2antrr 708 . . . . . . . . 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 738 . . . . . . . . 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 5900 . . . . . . . 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 734 . . . . . . . . 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 5900 . . . . . . . 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 16773 . . . . . . . 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 644 . . . . . . 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 736 . . . . . . . . 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 737 . . . . . . . . 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 739 . . . . . . . . 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 18408 . . . . . . . . . . 11  |-  ( ( D  e.  ( * Met `  X )  /\  w  e.  X  /\  z  e.  X
)  ->  ( w D z )  =  ( z D w ) )
2117, 10, 8, 20syl3anc 1185 . . . . . . . . . 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 735 . . . . . . . . . 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 4257 . . . . . . . . 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 18916 . . . . . . . 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 18916 . . . . . . . 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 4240 . . . . . . . . 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 4240 . . . . . . . . 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 3794 . . . . . . . 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 644 . . . . . . 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 4257 . . . . . 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 600 . . . . 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 2795 . . . 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 4241 . . . . . . 7  |-  ( s  =  r  ->  (
( z D w )  <  s  <->  ( z D w )  < 
r ) )
3433imbi1d 310 . . . . . 6  |-  ( s  =  r  ->  (
( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r )  <-> 
( ( z D w )  <  r  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) ) )
3534ralbidv 2731 . . . . 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 3058 . . . 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 644 . . 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 2804 . 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 445 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  D  e.  ( * Met `  X
) )
4012xrsxmet 18871 . . 3  |-  C  e.  ( * Met `  RR* )
4115, 16metcn 18604 . . 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 645 . 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 890 1  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712    C_ wss 3306   ifcif 3763   class class class wbr 4237    e. cmpt 4291   `'ccnv 4906   ran crn 4908   -->wf 5479   ` cfv 5483  (class class class)co 6110   supcsup 7474   0cc0 9021    +oocpnf 9148   RR*cxr 9150    < clt 9151    <_ cle 9152   RR+crp 10643    - ecxne 10738   + ecxad 10739   [,]cicc 10950   distcds 13569   RR* scxrs 13753   * Metcxmt 16717   MetOpencmopn 16722    Cn ccn 17319
This theorem is referenced by:  metdscn2  18918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-ec 6936  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-q 10606  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-icc 10954  df-fz 11075  df-seq 11355  df-exp 11414  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-plusg 13573  df-mulr 13574  df-tset 13579  df-ple 13580  df-ds 13582  df-topgen 13698  df-xrs 13757  df-psmet 16725  df-xmet 16726  df-bl 16728  df-mopn 16729  df-top 16994  df-bases 16996  df-topon 16997  df-cn 17322  df-cnp 17323
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