MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metdsf Structured version   Unicode version

Theorem metdsf 21857
Description: The distance from a point to a set is a nonnegative extended real number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
Hypothesis
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |-> inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )
)
Assertion
Ref Expression
metdsf  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
Distinct variable groups:    x, y, D    x, S, y    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem metdsf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simplll 767 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  D  e.  ( *Met `  X ) )
2 simplr 761 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  x  e.  X )
3 simplr 761 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  S  C_  X )
43sselda 3465 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  y  e.  X )
5 xmetcl 21338 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x D y )  e. 
RR* )
61, 2, 4, 5syl3anc 1265 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  (
x D y )  e.  RR* )
7 eqid 2423 . . . . . 6  |-  ( y  e.  S  |->  ( x D y ) )  =  ( y  e.  S  |->  ( x D y ) )
86, 7fmptd 6059 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  (
y  e.  S  |->  ( x D y ) ) : S --> RR* )
9 frn 5750 . . . . 5  |-  ( ( y  e.  S  |->  ( x D y ) ) : S --> RR*  ->  ran  ( y  e.  S  |->  ( x D y ) )  C_  RR* )
108, 9syl 17 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  ran  ( y  e.  S  |->  ( x D y ) )  C_  RR* )
11 infxrcl 11621 . . . 4  |-  ( ran  ( y  e.  S  |->  ( x D y ) )  C_  RR*  -> inf ( ran  ( y  e.  S  |->  ( x D y ) ) , 
RR* ,  <  )  e. 
RR* )
1210, 11syl 17 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  -> inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )  e.  RR* )
13 xmetge0 21351 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  0  <_  ( x D y ) )
141, 2, 4, 13syl3anc 1265 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  0  <_  ( x D y ) )
1514ralrimiva 2840 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  A. y  e.  S  0  <_  ( x D y ) )
16 ovex 6331 . . . . . . 7  |-  ( x D y )  e. 
_V
1716rgenw 2787 . . . . . 6  |-  A. y  e.  S  ( x D y )  e. 
_V
18 breq2 4425 . . . . . . 7  |-  ( z  =  ( x D y )  ->  (
0  <_  z  <->  0  <_  ( x D y ) ) )
197, 18ralrnmpt 6044 . . . . . 6  |-  ( A. y  e.  S  (
x D y )  e.  _V  ->  ( A. z  e.  ran  ( y  e.  S  |->  ( x D y ) ) 0  <_ 
z  <->  A. y  e.  S 
0  <_  ( x D y ) ) )
2017, 19ax-mp 5 . . . . 5  |-  ( A. z  e.  ran  ( y  e.  S  |->  ( x D y ) ) 0  <_  z  <->  A. y  e.  S  0  <_  ( x D y ) )
2115, 20sylibr 216 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  A. z  e.  ran  ( y  e.  S  |->  ( x D y ) ) 0  <_  z )
22 0xr 9689 . . . . 5  |-  0  e.  RR*
23 infxrgelb 11623 . . . . 5  |-  ( ( ran  ( y  e.  S  |->  ( x D y ) )  C_  RR* 
/\  0  e.  RR* )  ->  ( 0  <_ inf ( ran  ( y  e.  S  |->  ( x D y ) ) , 
RR* ,  <  )  <->  A. z  e.  ran  ( y  e.  S  |->  ( x D y ) ) 0  <_  z ) )
2410, 22, 23sylancl 667 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  (
0  <_ inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )  <->  A. z  e.  ran  ( y  e.  S  |->  ( x D y ) ) 0  <_  z ) )
2521, 24mpbird 236 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  0  <_ inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )
)
26 elxrge0 11743 . . 3  |-  (inf ( ran  ( y  e.  S  |->  ( x D y ) ) , 
RR* ,  <  )  e.  ( 0 [,] +oo ) 
<->  (inf ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )  e.  RR*  /\  0  <_ inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )
) )
2712, 25, 26sylanbrc 669 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  -> inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )  e.  ( 0 [,] +oo )
)
28 metdscn.f . 2  |-  F  =  ( x  e.  X  |-> inf ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  <  )
)
2927, 28fmptd 6059 1  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776   _Vcvv 3082    C_ wss 3437   class class class wbr 4421    |-> cmpt 4480   ran crn 4852   -->wf 5595   ` cfv 5599  (class class class)co 6303  infcinf 7959   0cc0 9541   +oocpnf 9674   RR*cxr 9676    < clt 9677    <_ cle 9678   [,]cicc 11640   *Metcxmt 18948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-inf 7961  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-2 10670  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-icc 11644  df-xmet 18956
This theorem is referenced by:  metds0  21859  metdstri  21860  metdsre  21862  metdseq0  21863  metdscnlem  21864  metdscn  21865  metnrmlem1a  21867  metnrmlem1  21868  lebnumlem1  21981
  Copyright terms: Public domain W3C validator