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Theorem metdsre 20448
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 3665 . . 3  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
2 metxmet 19928 . . . . . . . . 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 20443 . . . . . . . . 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 5578 . . . . . . 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 5863 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  ( 0 [,] +oo ) )
12 elxrge0 11413 . . . . . . . . . . 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 3375 . . . . . . . . . . 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 19926 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  X )  /\  z  e.  X  /\  w  e.  X )  ->  (
z D w )  e.  RR )
2015, 18, 10, 19syl3anc 1218 . . . . . . . . 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 20447 . . . . . . . . . 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 11165 . . . . . . . . 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 1219 . . . . . . . 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 2818 . . . . . 6  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  A. w  e.  X  ( F `  w )  e.  RR )
29 ffnfv 5888 . . . . . 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 1690 . . 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 1184 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 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2620   A.wral 2734    C_ wss 3347   (/)c0 3656   class class class wbr 4311    e. cmpt 4369   `'ccnv 4858   ran crn 4860    Fn wfn 5432   -->wf 5433   ` cfv 5437  (class class class)co 6110   supcsup 7709   RRcr 9300   0cc0 9301   +oocpnf 9434   RR*cxr 9436    < clt 9437    <_ cle 9438   [,]cicc 11322   *Metcxmt 17820   Metcme 17821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-po 4660  df-so 4661  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-er 7120  df-ec 7122  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-2 10399  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-icc 11326  df-psmet 17828  df-xmet 17829  df-met 17830  df-bl 17831
This theorem is referenced by:  metdscn2  20452  lebnumlem1  20552  lebnumlem3  20554
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