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.

Step	Hyp	Ref	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* , `' < ) )
4	3	metdsf 18352	. . . . . . . . 9 |- ( ( D e. ( * Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
5	2, 4	sylan 457	. . . . . . . 8 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
6	5	adantr 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 )
8	6, 7	syl 15	. . . . . 6 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F Fn X )
9	5	adantr 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 ) )
12	9, 10, 11	syl2anc 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 ) ) )
14	13	simplbi 446	. . . . . . . . . 10 |- ( ( F ` w ) e. ( 0 [,] +oo ) -> ( F ` w ) e. RR* )
15	12, 14	syl 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 )
18	17	sselda 3180	. . . . . . . . . . 11 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> z e. X )
19	18	adantrr 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 )
21	16, 19, 10, 20	syl3anc 1182	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( z D w ) e. RR )
22	13	simprbi 450	. . . . . . . . . 10 |- ( ( F ` w ) e. ( 0 [,] +oo ) -> 0 <_ ( F ` w ) )
23	12, 22	syl 15	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> 0 <_ ( F ` w ) )
24	3	metdsle 18356	. . . . . . . . . 10 |- ( ( ( D e. ( * Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) <_ ( z D w ) )
25	2, 24	sylanl1 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 )
27	15, 21, 23, 25, 26	syl22anc 1183	. . . . . . . 8 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) e. RR )
28	27	anassrs 629	. . . . . . 7 |- ( ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) /\ w e. X ) -> ( F ` w ) e. RR )
29	28	ralrimiva 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 ) )
31	8, 29, 30	sylanbrc 645	. . . . 5 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F : X --> RR )
32	31	ex 423	. . . 4 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( z e. S -> F : X --> RR ) )
33	32	exlimdv 1664	. . 3 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( E. z z e. S -> F : X --> RR ) )
34	1, 33	syl5bi 208	. 2 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( S =/= (/) -> F : X --> RR ) )
35	34	3impia 1148	1 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F : X --> RR )

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