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Theorem metdsre 21225
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 3799 . . 3  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
2 metxmet 20705 . . . . . . . . 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 21220 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,] +oo ) )
52, 4sylan 471 . . . . . . . 8  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  F : X --> ( 0 [,] +oo ) )
65adantr 465 . . . . . . 7  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  F : X --> ( 0 [,] +oo ) )
7 ffn 5737 . . . . . . 7  |-  ( F : X --> ( 0 [,] +oo )  ->  F  Fn  X )
86, 7syl 16 . . . . . 6  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  F  Fn  X )
95adantr 465 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  F : X --> ( 0 [,] +oo ) )
10 simprr 756 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  w  e.  X )
119, 10ffvelrnd 6033 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  ( 0 [,] +oo ) )
12 elxrge0 11641 . . . . . . . . . . 11  |-  ( ( F `  w )  e.  ( 0 [,] +oo )  <->  ( ( F `
 w )  e. 
RR*  /\  0  <_  ( F `  w ) ) )
1312simplbi 460 . . . . . . . . . 10  |-  ( ( F `  w )  e.  ( 0 [,] +oo )  ->  ( F `
 w )  e. 
RR* )
1411, 13syl 16 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  RR* )
15 simpll 753 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  D  e.  ( Met `  X ) )
16 simpr 461 . . . . . . . . . . . 12  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  S  C_  X )
1716sselda 3509 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  z  e.  X )
1817adantrr 716 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
z  e.  X )
19 metcl 20703 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  X )  /\  z  e.  X  /\  w  e.  X )  ->  (
z D w )  e.  RR )
2015, 18, 10, 19syl3anc 1228 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( z D w )  e.  RR )
2112simprbi 464 . . . . . . . . . 10  |-  ( ( F `  w )  e.  ( 0 [,] +oo )  ->  0  <_ 
( F `  w
) )
2211, 21syl 16 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
0  <_  ( F `  w ) )
233metdsle 21224 . . . . . . . . . 10  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  <_  ( z D w ) )
242, 23sylanl1 650 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  <_  ( z D w ) )
25 xrrege0 11387 . . . . . . . . 9  |-  ( ( ( ( F `  w )  e.  RR*  /\  ( z D w )  e.  RR )  /\  ( 0  <_ 
( F `  w
)  /\  ( F `  w )  <_  (
z D w ) ) )  ->  ( F `  w )  e.  RR )
2614, 20, 22, 24, 25syl22anc 1229 . . . . . . . 8  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  RR )
2726anassrs 648 . . . . . . 7  |-  ( ( ( ( D  e.  ( Met `  X
)  /\  S  C_  X
)  /\  z  e.  S )  /\  w  e.  X )  ->  ( F `  w )  e.  RR )
2827ralrimiva 2881 . . . . . 6  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  A. w  e.  X  ( F `  w )  e.  RR )
29 ffnfv 6058 . . . . . 6  |-  ( F : X --> RR  <->  ( F  Fn  X  /\  A. w  e.  X  ( F `  w )  e.  RR ) )
308, 28, 29sylanbrc 664 . . . . 5  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  F : X --> RR )
3130ex 434 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  (
z  e.  S  ->  F : X --> RR ) )
3231exlimdv 1700 . . 3  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  ( E. z  z  e.  S  ->  F : X --> RR ) )
331, 32syl5bi 217 . 2  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  ( S  =/=  (/)  ->  F : X
--> RR ) )
34333impia 1193 1  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X
--> RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2817    C_ wss 3481   (/)c0 3790   class class class wbr 4453    |-> cmpt 4511   `'ccnv 5004   ran crn 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   supcsup 7912   RRcr 9503   0cc0 9504   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641   [,]cicc 11544   *Metcxmt 18273   Metcme 18274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-ec 7325  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-icc 11548  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284
This theorem is referenced by:  metdscn2  21229  lebnumlem1  21329  lebnumlem3  21331
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