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Theorem metds0 18870
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 18868 . . . . . . . . 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 3335 . . . . . . . . 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 5863 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  e.  ( 0 [,]  +oo )
)
7 elxrge0 10998 . . . . . . . 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 10733 . . . . . 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 18869 . . . . . 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 18431 . . . . . . . 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 3674 . . . . . . 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 2647 . . . 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 9121 . . . . . 6  |-  0  e.  RR*
32 xrleloe 10727 . . . . . 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 2440 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 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   0cc0 8980    +oocpnf 9107   RR*cxr 9109    < clt 9110    <_ cle 9111   [,]cicc 10909   * Metcxmt 16676   ballcbl 16678
This theorem is referenced by:  metdsle  18872  metnrmlem1  18879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-2 10048  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-icc 10913  df-psmet 16684  df-xmet 16685  df-bl 16687
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