MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metdsre Unicode version

Theorem metdsre 18357
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 3464 . . 3  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
2 metxmet 17899 . . . . . . . . 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 18352 . . . . . . . . 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 5389 . . . . . . 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 5663 . . . . . . . . . . 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 10747 . . . . . . . . . . 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 3180 . . . . . . . . . . 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 17897 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  X )  /\  z  e.  X  /\  w  e.  X )  ->  (
z D w )  e.  RR )
2116, 19, 10, 20syl3anc 1182 . . . . . . . . 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 18356 . . . . . . . . . 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 10503 . . . . . . . . 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 1183 . . . . . . . 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 2626 . . . . . 6  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  A. w  e.  X  ( F `  w )  e.  RR )
30 ffnfv 5685 . . . . . 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 1664 . . 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 1148 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 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   [,]cicc 10659   * Metcxmt 16369   Metcme 16370
This theorem is referenced by:  metdscn2  18361  lebnumlem1  18459  lebnumlem3  18461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-ec 6662  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-xmet 16373  df-met 16374  df-bl 16375
  Copyright terms: Public domain W3C validator