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Theorem metdsge 21479
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 1001 . . . 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 21477 . . . 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 4468 . 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 1035 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  R  e.  RR* )  /\  w  e.  S )  ->  D  e.  ( *Met `  X ) )
71adantr 465 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  R  e.  RR* )  /\  w  e.  S )  ->  A  e.  X )
8 simpl2 1000 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  S  C_  X
)
98sselda 3499 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  R  e.  RR* )  /\  w  e.  S )  ->  w  e.  X )
10 xmetcl 20960 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  w  e.  X
)  ->  ( A D w )  e. 
RR* )
116, 7, 9, 10syl3anc 1228 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  R  e.  RR* )  /\  w  e.  S )  ->  ( A D w )  e. 
RR* )
12 oveq2 6304 . . . . . 6  |-  ( y  =  w  ->  ( A D y )  =  ( A D w ) )
1312cbvmptv 4548 . . . . 5  |-  ( y  e.  S  |->  ( A D y ) )  =  ( w  e.  S  |->  ( A D w ) )
1411, 13fmptd 6056 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( y  e.  S  |->  ( A D y ) ) : S --> RR* )
15 frn 5743 . . . 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 461 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  R  e.  RR* )
18 infmxrgelb 11551 . . 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 661 . 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 465 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  R  e.  RR* )  /\  w  e.  S )  ->  R  e.  RR* )
21 elbl2 21019 . . . . . . 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 1229 . . . . . 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 9670 . . . . . . 7  |-  ( ( ( A D w )  e.  RR*  /\  R  e.  RR* )  ->  (
( A D w )  <  R  <->  -.  R  <_  ( A D w ) ) )
2411, 20, 23syl2anc 661 . . . . . 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 253 . . . . 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 329 . . . 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 2893 . . 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 6324 . . . . 5  |-  ( A D w )  e. 
_V
2928rgenw 2818 . . . 4  |-  A. w  e.  S  ( A D w )  e. 
_V
30 breq2 4460 . . . . 5  |-  ( z  =  ( A D w )  ->  ( R  <_  z  <->  R  <_  ( A D w ) ) )
3113, 30ralrnmpt 6041 . . . 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 5 . . 3  |-  ( A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_  z  <->  A. w  e.  S  R  <_  ( A D w ) )
33 disj 3870 . . 3  |-  ( ( S  i^i  ( A ( ball `  D
) R ) )  =  (/)  <->  A. w  e.  S  -.  w  e.  ( A ( ball `  D
) R ) )
3427, 32, 333bitr4g 288 . 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 279 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   RR*cxr 9644    < clt 9645    <_ cle 9646   *Metcxmt 18530   ballcbl 18532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-psmet 18538  df-xmet 18539  df-bl 18541
This theorem is referenced by:  metds0  21480  metdstri  21481  metdseq0  21484  lebnumlem3  21589
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