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Theorem metdscn 20437
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 20429 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
3 iccssxr 11383 . . 3  |-  ( 0 [,] +oo )  C_  RR*
4 fss 5572 . . 3  |-  ( ( F : X --> ( 0 [,] +oo )  /\  ( 0 [,] +oo )  C_  RR* )  ->  F : X --> RR* )
52, 3, 4sylancl 662 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> RR* )
6 simprr 756 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR+ )
75ad2antrr 725 . . . . . . . . 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 759 . . . . . . . . 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 5849 . . . . . . . 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 755 . . . . . . . . 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 5849 . . . . . . . 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 17862 . . . . . . . 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 661 . . . . . . 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 757 . . . . . . . . 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 758 . . . . . . . . 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 760 . . . . . . . . 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 19927 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  X )  /\  w  e.  X  /\  z  e.  X
)  ->  ( w D z )  =  ( z D w ) )
2117, 10, 8, 20syl3anc 1218 . . . . . . . . . 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 756 . . . . . . . . . 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 4317 . . . . . . . . 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 20436 . . . . . . . 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 20436 . . . . . . . 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 4300 . . . . . . . . 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 4300 . . . . . . . . 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 3830 . . . . . . . 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 661 . . . . . . 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 4317 . . . . . 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 615 . . . . 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 2804 . . . 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 4301 . . . . . . 7  |-  ( s  =  r  ->  (
( z D w )  <  s  <->  ( z D w )  < 
r ) )
3433imbi1d 317 . . . . . 6  |-  ( s  =  r  ->  (
( ( z D w )  <  s  ->  ( ( F `  z ) C ( F `  w ) )  <  r )  <-> 
( ( z D w )  <  r  ->  ( ( F `  z ) C ( F `  w ) )  <  r ) ) )
3534ralbidv 2740 . . . . 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 3078 . . . 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 661 . . 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 2813 . 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 457 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  D  e.  ( *Met `  X
) )
4012xrsxmet 20391 . . 3  |-  C  e.  ( *Met `  RR* )
4115, 16metcn 20123 . . 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 662 . 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 913 1  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721    C_ wss 3333   ifcif 3796   class class class wbr 4297    e. cmpt 4355   `'ccnv 4844   ran crn 4846   -->wf 5419   ` cfv 5423  (class class class)co 6096   supcsup 7695   0cc0 9287   +oocpnf 9420   RR*cxr 9422    < clt 9423    <_ cle 9424   RR+crp 10996    -ecxne 11091   +ecxad 11092   [,]cicc 11308   distcds 14252   RR*scxrs 14443   *Metcxmt 17806   MetOpencmopn 17811    Cn ccn 18833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-ec 7108  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-icc 11312  df-fz 11443  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-tset 14262  df-ple 14263  df-ds 14265  df-topgen 14387  df-xrs 14445  df-psmet 17814  df-xmet 17815  df-bl 17817  df-mopn 17818  df-top 18508  df-bases 18510  df-topon 18511  df-cn 18836  df-cnp 18837
This theorem is referenced by:  metdscn2  20438
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