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Theorem metds0 21646
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 21644 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
323adant3 1017 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  F : X
--> ( 0 [,] +oo ) )
4 ssel2 3437 . . . . . . . . 9  |-  ( ( S  C_  X  /\  A  e.  S )  ->  A  e.  X )
543adant1 1015 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  A  e.  X )
63, 5ffvelrnd 6010 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  e.  ( 0 [,] +oo )
)
7 elxrge0 11683 . . . . . . . 8  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  <->  ( ( F `
 A )  e. 
RR*  /\  0  <_  ( F `  A ) ) )
87simplbi 458 . . . . . . 7  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  ( F `
 A )  e. 
RR* )
96, 8syl 17 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  e.  RR* )
10 xrleid 11409 . . . . . 6  |-  ( ( F `  A )  e.  RR*  ->  ( F `
 A )  <_ 
( F `  A
) )
119, 10syl 17 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  <_  ( F `  A )
)
12 simp1 997 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  D  e.  ( *Met `  X
) )
13 simp2 998 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  S  C_  X
)
141metdsge 21645 . . . . . 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 1233 . . . . 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 210 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( S  i^i  ( A ( ball `  D ) ( F `
 A ) ) )  =  (/) )
17 simpl3 1002 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  S )
1812adantr 463 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  D  e.  ( *Met `  X
) )
195adantr 463 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  X )
209adantr 463 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  ( F `  A )  e.  RR* )
21 simpr 459 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  0  <  ( F `  A ) )
22 xblcntr 21206 . . . . . . . 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 1234 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  ( A ( ball `  D
) ( F `  A ) ) )
24 inelcm 3824 . . . . . . 7  |-  ( ( A  e.  S  /\  A  e.  ( A
( ball `  D )
( F `  A
) ) )  -> 
( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )  =/=  (/) )
2517, 23, 24syl2anc 659 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  ( S  i^i  ( A ( ball `  D ) ( F `
 A ) ) )  =/=  (/) )
2625ex 432 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( 0  <  ( F `  A )  ->  ( S  i^i  ( A (
ball `  D )
( F `  A
) ) )  =/=  (/) ) )
2726necon2bd 2618 . . . 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 462 . . . . . 6  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  ->  0  <_ 
( F `  A
) )
306, 29syl 17 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  0  <_  ( F `  A ) )
31 0xr 9670 . . . . . 6  |-  0  e.  RR*
32 xrleloe 11403 . . . . . 6  |-  ( ( 0  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
3331, 9, 32sylancr 661 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( 0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
3430, 33mpbid 210 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) )
3534ord 375 . . 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 2410 1  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    i^i cin 3413    C_ wss 3414   (/)c0 3738   class class class wbr 4395    |-> cmpt 4453   `'ccnv 4822   ran crn 4824   -->wf 5565   ` cfv 5569  (class class class)co 6278   supcsup 7934   0cc0 9522   +oocpnf 9655   RR*cxr 9657    < clt 9658    <_ cle 9659   [,]cicc 11585   *Metcxmt 18723   ballcbl 18725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-2 10635  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-icc 11589  df-psmet 18731  df-xmet 18732  df-bl 18734
This theorem is referenced by:  metdsle  21648  metnrmlem1  21655
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