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Theorem metdscn 18457
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 18449 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
3 iccssxr 10821 . . 3  |-  ( 0 [,]  +oo )  C_  RR*
4 fss 5477 . . 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 5743 . . . . . . . . 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 5743 . . . . . . . . 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 16515 . . . . . . . 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 18008 . . . . . . . . . . 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 4122 . . . . . . . . 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 18456 . . . . . . . 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 18456 . . . . . . . 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 4105 . . . . . . . . 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 4105 . . . . . . . . 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 3672 . . . . . . . 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 4122 . . . . . 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 2702 . . . 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 4106 . . . . . . 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 2639 . . . . 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 2960 . . . 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 2711 . 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 18411 . . 3  |-  C  e.  ( * Met `  RR* )
4317, 18metcn 18185 . . 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 1642    e. wcel 1710   A.wral 2619   E.wrex 2620    C_ wss 3228   ifcif 3641   class class class wbr 4102    e. cmpt 4156   `'ccnv 4767   ran crn 4769   -->wf 5330   ` cfv 5334  (class class class)co 5942   supcsup 7280   0cc0 8824    +oocpnf 8951   RR*cxr 8953    < clt 8954    <_ cle 8955   RR+crp 10443    - ecxne 10538   + ecxad 10539   [,]cicc 10748   distcds 13308   RR* scxrs 13492   * Metcxmt 16462   MetOpencmopn 16467    Cn ccn 17054
This theorem is referenced by:  metdscn2  18458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-ec 6746  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-icc 10752  df-fz 10872  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-plusg 13312  df-mulr 13313  df-tset 13318  df-ple 13319  df-ds 13321  df-topgen 13437  df-xrs 13496  df-xmet 16469  df-bl 16471  df-mopn 16472  df-top 16736  df-bases 16738  df-topon 16739  df-cn 17057  df-cnp 17058
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