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Theorem metds0 20561
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 20559 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
323adant3 1008 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  F : X
--> ( 0 [,] +oo ) )
4 ssel2 3462 . . . . . . . . 9  |-  ( ( S  C_  X  /\  A  e.  S )  ->  A  e.  X )
543adant1 1006 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  A  e.  X )
63, 5ffvelrnd 5956 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  e.  ( 0 [,] +oo )
)
7 elxrge0 11514 . . . . . . . 8  |-  ( ( F `  A )  e.  ( 0 [,] +oo )  <->  ( ( F `
 A )  e. 
RR*  /\  0  <_  ( F `  A ) ) )
87simplbi 460 . . . . . . 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 11241 . . . . . 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 988 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  D  e.  ( *Met `  X
) )
13 simp2 989 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  S  C_  X
)
141metdsge 20560 . . . . . 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 1222 . . . . 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 993 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  S )
1812adantr 465 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  D  e.  ( *Met `  X
) )
195adantr 465 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  X )
209adantr 465 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  ( F `  A )  e.  RR* )
21 simpr 461 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  0  <  ( F `  A ) )
22 xblcntr 20121 . . . . . . . 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 1223 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  A  e.  ( A ( ball `  D
) ( F `  A ) ) )
24 inelcm 3844 . . . . . . 7  |-  ( ( A  e.  S  /\  A  e.  ( A
( ball `  D )
( F `  A
) ) )  -> 
( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )  =/=  (/) )
2517, 23, 24syl2anc 661 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  S
)  /\  0  <  ( F `  A ) )  ->  ( S  i^i  ( A ( ball `  D ) ( F `
 A ) ) )  =/=  (/) )
2625ex 434 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( 0  <  ( F `  A )  ->  ( S  i^i  ( A (
ball `  D )
( F `  A
) ) )  =/=  (/) ) )
2726necon2bd 2667 . . . 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 464 . . . . . 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 9544 . . . . . 6  |-  0  e.  RR*
32 xrleloe 11235 . . . . . 6  |-  ( ( 0  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
3331, 9, 32sylancr 663 . . . . 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 377 . . 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 2462 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 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648    i^i cin 3438    C_ wss 3439   (/)c0 3748   class class class wbr 4403    |-> cmpt 4461   `'ccnv 4950   ran crn 4952   -->wf 5525   ` cfv 5529  (class class class)co 6203   supcsup 7804   0cc0 9396   +oocpnf 9529   RR*cxr 9531    < clt 9532    <_ cle 9533   [,]cicc 11417   *Metcxmt 17929   ballcbl 17931
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-2 10494  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-icc 11421  df-psmet 17937  df-xmet 17938  df-bl 17940
This theorem is referenced by:  metdsle  20563  metnrmlem1  20570
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