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Theorem metdsf 21478
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.)
Hypothesis
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |->  sup ( 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 759 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  D  e.  ( *Met `  X ) )
2 simplr 755 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  x  e.  X )
3 simplr 755 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  S  C_  X )
43sselda 3499 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  y  e.  X )
5 xmetcl 20960 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x D y )  e. 
RR* )
61, 2, 4, 5syl3anc 1228 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  (
x D y )  e.  RR* )
7 eqid 2457 . . . . . 6  |-  ( y  e.  S  |->  ( x D y ) )  =  ( y  e.  S  |->  ( x D y ) )
86, 7fmptd 6056 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  (
y  e.  S  |->  ( x D y ) ) : S --> RR* )
9 frn 5743 . . . . 5  |-  ( ( y  e.  S  |->  ( x D y ) ) : S --> RR*  ->  ran  ( y  e.  S  |->  ( x D y ) )  C_  RR* )
108, 9syl 16 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  ran  ( y  e.  S  |->  ( x D y ) )  C_  RR* )
11 infmxrcl 11533 . . . 4  |-  ( ran  ( y  e.  S  |->  ( x D y ) )  C_  RR*  ->  sup ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  )  e.  RR* )
1210, 11syl 16 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  sup ( ran  ( y  e.  S  |->  ( x D y ) ) , 
RR* ,  `'  <  )  e.  RR* )
13 xmetge0 20973 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  0  <_  ( x D y ) )
141, 2, 4, 13syl3anc 1228 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  0  <_  ( x D y ) )
1514ralrimiva 2871 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  A. y  e.  S  0  <_  ( x D y ) )
16 ovex 6324 . . . . . . 7  |-  ( x D y )  e. 
_V
1716rgenw 2818 . . . . . 6  |-  A. y  e.  S  ( x D y )  e. 
_V
18 breq2 4460 . . . . . . 7  |-  ( z  =  ( x D y )  ->  (
0  <_  z  <->  0  <_  ( x D y ) ) )
197, 18ralrnmpt 6041 . . . . . 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 212 . . . 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 9657 . . . . 5  |-  0  e.  RR*
23 infmxrgelb 11551 . . . . 5  |-  ( ( ran  ( y  e.  S  |->  ( x D y ) )  C_  RR* 
/\  0  e.  RR* )  ->  ( 0  <_  sup ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  )  <->  A. z  e.  ran  ( y  e.  S  |->  ( x D y ) ) 0  <_ 
z ) )
2410, 22, 23sylancl 662 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  (
0  <_  sup ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  )  <->  A. z  e.  ran  ( y  e.  S  |->  ( x D y ) ) 0  <_  z ) )
2521, 24mpbird 232 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  0  <_  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
26 elxrge0 11654 . . 3  |-  ( sup ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  )  e.  ( 0 [,] +oo )  <->  ( sup ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  )  e.  RR*  /\  0  <_  sup ( ran  ( y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) ) )
2712, 25, 26sylanbrc 664 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  sup ( ran  ( y  e.  S  |->  ( x D y ) ) , 
RR* ,  `'  <  )  e.  ( 0 [,] +oo ) )
28 metdscn.f . 2  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
2927, 28fmptd 6056 1  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   0cc0 9509   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   [,]cicc 11557   *Metcxmt 18530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-icc 11561  df-xmet 18539
This theorem is referenced by:  metds0  21480  metdstri  21481  metdsre  21483  metdseq0  21484  metdscnlem  21485  metdscn  21486  metnrmlem1a  21488  metnrmlem1  21489  lebnumlem1  21587
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