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Theorem metdsf 20549
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 757 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  D  e.  ( *Met `  X ) )
2 simplr 754 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  x  e.  X )
3 simplr 754 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  S  C_  X )
43sselda 3457 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  y  e.  X )
5 xmetcl 20031 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x D y )  e. 
RR* )
61, 2, 4, 5syl3anc 1219 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  (
x D y )  e.  RR* )
7 eqid 2451 . . . . . 6  |-  ( y  e.  S  |->  ( x D y ) )  =  ( y  e.  S  |->  ( x D y ) )
86, 7fmptd 5969 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  (
y  e.  S  |->  ( x D y ) ) : S --> RR* )
9 frn 5666 . . . . 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 11383 . . . 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 20044 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  0  <_  ( x D y ) )
141, 2, 4, 13syl3anc 1219 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X )  /\  x  e.  X )  /\  y  e.  S )  ->  0  <_  ( x D y ) )
1514ralrimiva 2825 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  x  e.  X )  ->  A. y  e.  S  0  <_  ( x D y ) )
16 ovex 6218 . . . . . . 7  |-  ( x D y )  e. 
_V
1716rgenw 2894 . . . . . 6  |-  A. y  e.  S  ( x D y )  e. 
_V
18 breq2 4397 . . . . . . 7  |-  ( z  =  ( x D y )  ->  (
0  <_  z  <->  0  <_  ( x D y ) ) )
197, 18ralrnmpt 5954 . . . . . 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 9534 . . . . 5  |-  0  e.  RR*
23 infmxrgelb 11401 . . . . 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 11504 . . 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 5969 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 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071    C_ wss 3429   class class class wbr 4393    |-> cmpt 4451   `'ccnv 4940   ran crn 4942   -->wf 5515   ` cfv 5519  (class class class)co 6193   supcsup 7794   0cc0 9386   +oocpnf 9519   RR*cxr 9521    < clt 9522    <_ cle 9523   [,]cicc 11407   *Metcxmt 17919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-2 10484  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-icc 11411  df-xmet 17928
This theorem is referenced by:  metds0  20551  metdstri  20552  metdsre  20554  metdseq0  20555  metdscnlem  20556  metdscn  20557  metnrmlem1a  20559  metnrmlem1  20560  lebnumlem1  20658
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