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Theorem metdsge 18871
Description: The distance from the point  A to the set  S is greater than  R iff the  R-ball around  A misses  S. (Contributed 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
metdsge  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( R  <_  ( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) R ) )  =  (/) ) )
Distinct variable groups:    x, y, A    x, D, y    x, S, y    x, X, y
Allowed substitution hints:    R( x, y)    F( x, y)

Proof of Theorem metdsge
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 962 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  A  e.  X )
2 metdscn.f . . . . 5  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
32metdsval 18869 . . . 4  |-  ( A  e.  X  ->  ( F `  A )  =  sup ( ran  (
y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) )
41, 3syl 16 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( F `  A )  =  sup ( ran  ( y  e.  S  |->  ( A D y ) ) , 
RR* ,  `'  <  ) )
54breq2d 4216 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( R  <_  ( F `  A
)  <->  R  <_  sup ( ran  ( y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) ) )
6 simpll1 996 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  D  e.  ( * Met `  X
) )
71adantr 452 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  A  e.  X )
8 simpl2 961 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  S  C_  X
)
98sselda 3340 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  w  e.  X )
10 xmetcl 18353 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  w  e.  X
)  ->  ( A D w )  e. 
RR* )
116, 7, 9, 10syl3anc 1184 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  ( A D w )  e. 
RR* )
12 oveq2 6081 . . . . . 6  |-  ( y  =  w  ->  ( A D y )  =  ( A D w ) )
1312cbvmptv 4292 . . . . 5  |-  ( y  e.  S  |->  ( A D y ) )  =  ( w  e.  S  |->  ( A D w ) )
1411, 13fmptd 5885 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( y  e.  S  |->  ( A D y ) ) : S --> RR* )
15 frn 5589 . . . 4  |-  ( ( y  e.  S  |->  ( A D y ) ) : S --> RR*  ->  ran  ( y  e.  S  |->  ( A D y ) )  C_  RR* )
1614, 15syl 16 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ran  ( y  e.  S  |->  ( A D y ) ) 
C_  RR* )
17 simpr 448 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  R  e.  RR* )
18 infmxrgelb 10905 . . 3  |-  ( ( ran  ( y  e.  S  |->  ( A D y ) )  C_  RR* 
/\  R  e.  RR* )  ->  ( R  <_  sup ( ran  ( y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  )  <->  A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_ 
z ) )
1916, 17, 18syl2anc 643 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( R  <_  sup ( ran  (
y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  )  <->  A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_ 
z ) )
2017adantr 452 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  R  e.  RR* )
21 elbl2 18412 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  R  e.  RR* )  /\  ( A  e.  X  /\  w  e.  X ) )  -> 
( w  e.  ( A ( ball `  D
) R )  <->  ( A D w )  < 
R ) )
226, 20, 7, 9, 21syl22anc 1185 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  (
w  e.  ( A ( ball `  D
) R )  <->  ( A D w )  < 
R ) )
23 xrltnle 9136 . . . . . . 7  |-  ( ( ( A D w )  e.  RR*  /\  R  e.  RR* )  ->  (
( A D w )  <  R  <->  -.  R  <_  ( A D w ) ) )
2411, 20, 23syl2anc 643 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  (
( A D w )  <  R  <->  -.  R  <_  ( A D w ) ) )
2522, 24bitrd 245 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  (
w  e.  ( A ( ball `  D
) R )  <->  -.  R  <_  ( A D w ) ) )
2625con2bid 320 . . . 4  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  ( R  <_  ( A D w )  <->  -.  w  e.  ( A ( ball `  D ) R ) ) )
2726ralbidva 2713 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( A. w  e.  S  R  <_  ( A D w )  <->  A. w  e.  S  -.  w  e.  ( A ( ball `  D
) R ) ) )
28 ovex 6098 . . . . 5  |-  ( A D w )  e. 
_V
2928rgenw 2765 . . . 4  |-  A. w  e.  S  ( A D w )  e. 
_V
30 breq2 4208 . . . . 5  |-  ( z  =  ( A D w )  ->  ( R  <_  z  <->  R  <_  ( A D w ) ) )
3113, 30ralrnmpt 5870 . . . 4  |-  ( A. w  e.  S  ( A D w )  e. 
_V  ->  ( A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_  z  <->  A. w  e.  S  R  <_  ( A D w ) ) )
3229, 31ax-mp 8 . . 3  |-  ( A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_  z  <->  A. w  e.  S  R  <_  ( A D w ) )
33 disj 3660 . . 3  |-  ( ( S  i^i  ( A ( ball `  D
) R ) )  =  (/)  <->  A. w  e.  S  -.  w  e.  ( A ( ball `  D
) R ) )
3427, 32, 333bitr4g 280 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_  z  <->  ( S  i^i  ( A ( ball `  D ) R ) )  =  (/) ) )
355, 19, 343bitrd 271 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( R  <_  ( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) R ) )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    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   RR*cxr 9111    < clt 9112    <_ cle 9113   * Metcxmt 16678   ballcbl 16680
This theorem is referenced by:  metds0  18872  metdstri  18873  metdseq0  18876  lebnumlem3  18980
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 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-psmet 16686  df-xmet 16687  df-bl 16689
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