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Theorem metdscn 21797
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 21789 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
3 iccssxr 11706 . . 3  |-  ( 0 [,] +oo )  C_  RR*
4 fss 5745 . . 3  |-  ( ( F : X --> ( 0 [,] +oo )  /\  ( 0 [,] +oo )  C_  RR* )  ->  F : X --> RR* )
52, 3, 4sylancl 666 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> RR* )
6 simprr 764 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR+ )
75ad2antrr 730 . . . . . . . . 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 768 . . . . . . . . 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 6029 . . . . . . . 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 762 . . . . . . . . 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 6029 . . . . . . . 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 18940 . . . . . . . 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 665 . . . . . . 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 766 . . . . . . . . 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 767 . . . . . . . . 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 769 . . . . . . . . 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 21286 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  X )  /\  w  e.  X  /\  z  e.  X
)  ->  ( w D z )  =  ( z D w ) )
2117, 10, 8, 20syl3anc 1264 . . . . . . . . . 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 764 . . . . . . . . . 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 4437 . . . . . . . . 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 21796 . . . . . . . 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 21796 . . . . . . . 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 4420 . . . . . . . . 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 4420 . . . . . . . . 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 3942 . . . . . . . 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 665 . . . . . . 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 4437 . . . . . 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 618 . . . . 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 2837 . . . 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 4421 . . . . . . 7  |-  ( s  =  r  ->  (
( z D w )  <  s  <->  ( z D w )  < 
r ) )
3433imbi1d 318 . . . . . 6  |-  ( s  =  r  ->  (
( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r )  <-> 
( ( z D w )  <  r  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) ) )
3534ralbidv 2862 . . . . 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 3179 . . . 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 665 . . 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 2844 . 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 458 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  D  e.  ( *Met `  X
) )
4012xrsxmet 21751 . . 3  |-  C  e.  ( *Met `  RR* )
4115, 16metcn 21482 . . 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 666 . 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 930 1  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774    C_ wss 3433   ifcif 3906   class class class wbr 4417    |-> cmpt 4475   `'ccnv 4844   ran crn 4846   -->wf 5588   ` cfv 5592  (class class class)co 6296   supcsup 7951   0cc0 9528   +oocpnf 9661   RR*cxr 9663    < clt 9664    <_ cle 9665   RR+crp 11291    -ecxne 11395   +ecxad 11396   [,]cicc 11627   distcds 15151   RR*scxrs 15350   *Metcxmt 18883   MetOpencmopn 18888    Cn ccn 20164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-ec 7364  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-icc 11631  df-fz 11772  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-plusg 15155  df-mulr 15156  df-tset 15161  df-ple 15162  df-ds 15164  df-topgen 15294  df-xrs 15352  df-psmet 18890  df-xmet 18891  df-bl 18893  df-mopn 18894  df-top 19845  df-bases 19846  df-topon 19847  df-cn 20167  df-cnp 20168
This theorem is referenced by:  metdscn2  21798
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