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Theorem metds0 18744
Description: If a point is in a set, its distance to the set is zero. (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
metds0  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
Distinct variable groups:    x, y, A    x, D, y    x, S, y    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem metds0
StepHypRef Expression
1 metdscn.f . . . . . . . . . 10  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
21metdsf 18742 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
323adant3 977 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  F : X
--> ( 0 [,]  +oo ) )
4 ssel2 3279 . . . . . . . . 9  |-  ( ( S  C_  X  /\  A  e.  S )  ->  A  e.  X )
543adant1 975 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  A  e.  X )
63, 5ffvelrnd 5803 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  e.  ( 0 [,]  +oo )
)
7 elxrge0 10933 . . . . . . . 8  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  A )  e.  RR*  /\  0  <_ 
( F `  A
) ) )
87simplbi 447 . . . . . . 7  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  ( F `  A )  e.  RR* )
96, 8syl 16 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  e.  RR* )
10 xrleid 10668 . . . . . 6  |-  ( ( F `  A )  e.  RR*  ->  ( F `
 A )  <_ 
( F `  A
) )
119, 10syl 16 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  <_  ( F `  A )
)
12 simp1 957 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  D  e.  ( * Met `  X
) )
13 simp2 958 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  S  C_  X
)
141metdsge 18743 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  e.  RR* )  ->  ( ( F `
 A )  <_ 
( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )  =  (/) ) )
1512, 13, 5, 9, 14syl31anc 1187 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( ( F `  A )  <_  ( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )  =  (/) ) )
1611, 15mpbid 202 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( S  i^i  ( A ( ball `  D ) ( F `
 A ) ) )  =  (/) )
17 simpl3 962 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  S )
1812adantr 452 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  D  e.  ( * Met `  X
) )
195adantr 452 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  X )
209adantr 452 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  ( F `  A )  e.  RR* )
21 simpr 448 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  0  <  ( F `  A ) )
22 xblcntr 18330 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  ( ( F `  A )  e.  RR*  /\  0  <  ( F `
 A ) ) )  ->  A  e.  ( A ( ball `  D
) ( F `  A ) ) )
2318, 19, 20, 21, 22syl112anc 1188 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  ( A ( ball `  D
) ( F `  A ) ) )
24 inelcm 3618 . . . . . . 7  |-  ( ( A  e.  S  /\  A  e.  ( A
( ball `  D )
( F `  A
) ) )  -> 
( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )  =/=  (/) )
2517, 23, 24syl2anc 643 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  ( S  i^i  ( A ( ball `  D ) ( F `
 A ) ) )  =/=  (/) )
2625ex 424 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( 0  <  ( F `  A )  ->  ( S  i^i  ( A (
ball `  D )
( F `  A
) ) )  =/=  (/) ) )
2726necon2bd 2592 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( ( S  i^i  ( A (
ball `  D )
( F `  A
) ) )  =  (/)  ->  -.  0  <  ( F `  A ) ) )
2816, 27mpd 15 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  -.  0  <  ( F `  A
) )
297simprbi 451 . . . . . 6  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  A
) )
306, 29syl 16 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  0  <_  ( F `  A ) )
31 0xr 9057 . . . . . 6  |-  0  e.  RR*
32 xrleloe 10662 . . . . . 6  |-  ( ( 0  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
3331, 9, 32sylancr 645 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( 0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
3430, 33mpbid 202 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) )
3534ord 367 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( -.  0  <  ( F `  A )  ->  0  =  ( F `  A ) ) )
3628, 35mpd 15 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  0  =  ( F `  A ) )
3736eqcomd 2385 1  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543    i^i cin 3255    C_ wss 3256   (/)c0 3564   class class class wbr 4146    e. cmpt 4200   `'ccnv 4810   ran crn 4812   -->wf 5383   ` cfv 5387  (class class class)co 6013   supcsup 7373   0cc0 8916    +oocpnf 9043   RR*cxr 9045    < clt 9046    <_ cle 9047   [,]cicc 10844   * Metcxmt 16605   ballcbl 16607
This theorem is referenced by:  metdsle  18746  metnrmlem1  18753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-icc 10848  df-xmet 16612  df-bl 16614
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