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Theorem metdsre 18844
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 3605 . . 3  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
2 metxmet 18325 . . . . . . . . 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 18839 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
52, 4sylan 458 . . . . . . . 8  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  F : X --> ( 0 [,] 
+oo ) )
65adantr 452 . . . . . . 7  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  F : X --> ( 0 [,] 
+oo ) )
7 ffn 5558 . . . . . . 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 452 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  F : X --> ( 0 [,]  +oo ) )
10 simprr 734 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  w  e.  X )
119, 10ffvelrnd 5838 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  ( 0 [,]  +oo ) )
12 elxrge0 10972 . . . . . . . . . . 11  |-  ( ( F `  w )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  w )  e.  RR*  /\  0  <_ 
( F `  w
) ) )
1312simplbi 447 . . . . . . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  ->  D  e.  ( Met `  X ) )
16 simpr 448 . . . . . . . . . . . 12  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  S  C_  X )
1716sselda 3316 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  z  e.  X )
1817adantrr 698 . . . . . . . . . 10  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
z  e.  X )
19 metcl 18323 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  X )  /\  z  e.  X  /\  w  e.  X )  ->  (
z D w )  e.  RR )
2015, 18, 10, 19syl3anc 1184 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( z D w )  e.  RR )
2112simprbi 451 . . . . . . . . . 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 18843 . . . . . . . . . 10  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  <_  ( z D w ) )
242, 23sylanl1 632 . . . . . . . . 9  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  <_  ( z D w ) )
25 xrrege0 10726 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( z  e.  S  /\  w  e.  X ) )  -> 
( F `  w
)  e.  RR )
2726anassrs 630 . . . . . . 7  |-  ( ( ( ( D  e.  ( Met `  X
)  /\  S  C_  X
)  /\  z  e.  S )  /\  w  e.  X )  ->  ( F `  w )  e.  RR )
2827ralrimiva 2757 . . . . . 6  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  A. w  e.  X  ( F `  w )  e.  RR )
29 ffnfv 5861 . . . . . 6  |-  ( F : X --> RR  <->  ( F  Fn  X  /\  A. w  e.  X  ( F `  w )  e.  RR ) )
308, 28, 29sylanbrc 646 . . . . 5  |-  ( ( ( D  e.  ( Met `  X )  /\  S  C_  X
)  /\  z  e.  S )  ->  F : X --> RR )
3130ex 424 . . . 4  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  (
z  e.  S  ->  F : X --> RR ) )
3231exlimdv 1643 . . 3  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  ( E. z  z  e.  S  ->  F : X --> RR ) )
331, 32syl5bi 209 . 2  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X )  ->  ( S  =/=  (/)  ->  F : X
--> RR ) )
34333impia 1150 1  |-  ( ( D  e.  ( Met `  X )  /\  S  C_  X  /\  S  =/=  (/) )  ->  F : X
--> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674    C_ wss 3288   (/)c0 3596   class class class wbr 4180    e. cmpt 4234   `'ccnv 4844   ran crn 4846    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048   supcsup 7411   RRcr 8953   0cc0 8954    +oocpnf 9081   RR*cxr 9083    < clt 9084    <_ cle 9085   [,]cicc 10883   * Metcxmt 16649   Metcme 16650
This theorem is referenced by:  metdscn2  18848  lebnumlem1  18947  lebnumlem3  18949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-er 6872  df-ec 6874  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-2 10022  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-icc 10887  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660
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