Theorem metdsre 22656

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 |-> inf ( 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.

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( S =/= (/) <-> E. z z e. S )
2		metxmet 22139	. . . . . . . . 9 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
3		metdscn.f	. . . . . . . . . 10 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
4	3	metdsf 22651	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
5	2, 4	sylan 488	. . . . . . . 8 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
6	5	adantr 481	. . . . . . 7 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F : X --> ( 0 [,] +oo ) )
7		ffn 6045	. . . . . . 7 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
8	6, 7	syl 17	. . . . . 6 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F Fn X )
9	5	adantr 481	. . . . . . . . . . 11 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> F : X --> ( 0 [,] +oo ) )
10		simprr 796	. . . . . . . . . . 11 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> w e. X )
11	9, 10	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) e. ( 0 [,] +oo ) )
12		elxrge0 12281	. . . . . . . . . . 11 |- ( ( F ` w ) e. ( 0 [,] +oo ) <-> ( ( F ` w ) e. RR* /\ 0 <_ ( F ` w ) ) )
13	12	simplbi 476	. . . . . . . . . 10 |- ( ( F ` w ) e. ( 0 [,] +oo ) -> ( F ` w ) e. RR* )
14	11, 13	syl 17	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) e. RR* )
15		simpll 790	. . . . . . . . . 10 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> D e. ( Met ` X ) )
16		simpr 477	. . . . . . . . . . . 12 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> S C_ X )
17	16	sselda 3603	. . . . . . . . . . 11 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> z e. X )
18	17	adantrr 753	. . . . . . . . . 10 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> z e. X )
19		metcl 22137	. . . . . . . . . 10 |- ( ( D e. ( Met ` X ) /\ z e. X /\ w e. X ) -> ( z D w ) e. RR )
20	15, 18, 10, 19	syl3anc 1326	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( z D w ) e. RR )
21	12	simprbi 480	. . . . . . . . . 10 |- ( ( F ` w ) e. ( 0 [,] +oo ) -> 0 <_ ( F ` w ) )
22	11, 21	syl 17	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> 0 <_ ( F ` w ) )
23	3	metdsle 22655	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) <_ ( z D w ) )
24	2, 23	sylanl1 682	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) <_ ( z D w ) )
25		xrrege0 12005	. . . . . . . . 9 |- ( ( ( ( F ` w ) e. RR* /\ ( z D w ) e. RR ) /\ ( 0 <_ ( F ` w ) /\ ( F ` w ) <_ ( z D w ) ) ) -> ( F ` w ) e. RR )
26	14, 20, 22, 24, 25	syl22anc 1327	. . . . . . . 8 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) e. RR )
27	26	anassrs 680	. . . . . . 7 |- ( ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) /\ w e. X ) -> ( F ` w ) e. RR )
28	27	ralrimiva 2966	. . . . . 6 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> A. w e. X ( F ` w ) e. RR )
29		ffnfv 6388	. . . . . 6 |- ( F : X --> RR <-> ( F Fn X /\ A. w e. X ( F ` w ) e. RR ) )
30	8, 28, 29	sylanbrc 698	. . . . 5 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F : X --> RR )
31	30	ex 450	. . . 4 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( z e. S -> F : X --> RR ) )
32	31	exlimdv 1861	. . 3 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( E. z z e. S -> F : X --> RR ) )
33	1, 32	syl5bi 232	. 2 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( S =/= (/) -> F : X --> RR ) )
34	33	3impia 1261	1 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F : X --> RR )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,]cicc 12178   *Metcxmt 19731   Metcme 19732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-ec 7744  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741

This theorem is referenced by:  metdscn2  22660  lebnumlem1  22760  lebnumlem3  22762