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Theorem metdsre 18571
Description: The distance from a point to a nonempty set in a proper metric space is a real number. (Contributed by Mario Carneiro, 5-Sep-2015.)
Hypothesis
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
Assertion
Ref Expression
metdsre  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X
--> RR )
Distinct variable groups:    x, y, D    x, S, y    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem metdsre
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3552 . . 3  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
2 metxmet 18112 . . . . . . . . 9  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
3 metdscn.f . . . . . . . . . 10  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
43metdsf 18566 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
52, 4sylan 457 . . . . . . . 8  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  F : X --> ( 0 [,] 
+oo ) )
65adantr 451 . . . . . . 7  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  F : X --> ( 0 [,] 
+oo ) )
7 ffn 5495 . . . . . . 7  |-  ( F : X --> ( 0 [,]  +oo )  ->  F  Fn  X )
86, 7syl 15 . . . . . 6  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  F  Fn  X )
95adantr 451 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  F : X --> ( 0 [,]  +oo ) )
10 simprr 733 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  w  e.  X )
11 ffvelrn 5770 . . . . . . . . . . 11  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  w  e.  X )  ->  ( F `  w )  e.  ( 0 [,]  +oo ) )
129, 10, 11syl2anc 642 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  ( 0 [,]  +oo ) )
13 elxrge0 10900 . . . . . . . . . . 11  |-  ( ( F `  w )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  w )  e.  RR*  /\  0  <_ 
( F `  w
) ) )
1413simplbi 446 . . . . . . . . . 10  |-  ( ( F `  w )  e.  ( 0 [,] 
+oo )  ->  ( F `  w )  e.  RR* )
1512, 14syl 15 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  RR* )
16 simpll 730 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  D  e.  ( Met `  X ) )
17 simpr 447 . . . . . . . . . . . 12  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  S  C_  X )
1817sselda 3266 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  z  e.  X )
1918adantrr 697 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
z  e.  X )
20 metcl 18110 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  X )  /\  z  e.  X  /\  w  e.  X )  ->  (
z D w )  e.  RR )
2116, 19, 10, 20syl3anc 1183 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( z D w )  e.  RR )
2213simprbi 450 . . . . . . . . . 10  |-  ( ( F `  w )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  w
) )
2312, 22syl 15 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
0  <_  ( F `  w ) )
243metdsle 18570 . . . . . . . . . 10  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  <_  ( z D w ) )
252, 24sylanl1 631 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  <_  ( z D w ) )
26 xrrege0 10655 . . . . . . . . 9  |-  ( ( ( ( F `  w )  e.  RR*  /\  ( z D w )  e.  RR )  /\  ( 0  <_ 
( F `  w
)  /\  ( F `  w )  <_  (
z D w ) ) )  ->  ( F `  w )  e.  RR )
2715, 21, 23, 25, 26syl22anc 1184 . . . . . . . 8  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  RR )
2827anassrs 629 . . . . . . 7  |-  ( ( ( ( D  e.  ( Met `  X
)  /\  S  C_  X
)  /\  z  e.  S )  /\  w  e.  X )  ->  ( F `  w )  e.  RR )
2928ralrimiva 2711 . . . . . 6  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  A. w  e.  X  ( F `  w )  e.  RR )
30 ffnfv 5796 . . . . . 6  |-  ( F : X --> RR  <->  ( F  Fn  X  /\  A. w  e.  X  ( F `  w )  e.  RR ) )
318, 29, 30sylanbrc 645 . . . . 5  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  F : X --> RR )
3231ex 423 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  (
z  e.  S  ->  F : X --> RR ) )
3332exlimdv 1641 . . 3  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  ( E. z  z  e.  S  ->  F : X --> RR ) )
341, 33syl5bi 208 . 2  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  ( S  =/=  (/)  ->  F : X
--> RR ) )
35343impia 1149 1  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X
--> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935   E.wex 1546    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628    C_ wss 3238   (/)c0 3543   class class class wbr 4125    e. cmpt 4179   `'ccnv 4791   ran crn 4793    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981   supcsup 7340   RRcr 8883   0cc0 8884    +oocpnf 9011   RR*cxr 9013    < clt 9014    <_ cle 9015   [,]cicc 10812   * Metcxmt 16579   Metcme 16580
This theorem is referenced by:  metdscn2  18575  lebnumlem1  18674  lebnumlem3  18676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-er 6802  df-ec 6804  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-2 9951  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-icc 10816  df-xmet 16586  df-met 16587  df-bl 16588
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