Theorem metdsre 18888

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 3639	. . 3 |- ( S =/= (/) <-> E. z z e. S )
2		metxmet 18369	. . . . . . . . 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 18883	. . . . . . . . 9 |- ( ( D e. ( * Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
5	2, 4	sylan 459	. . . . . . . 8 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
6	5	adantr 453	. . . . . . 7 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F : X --> ( 0 [,] +oo ) )
7		ffn 5594	. . . . . . 7 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
8	6, 7	syl 16	. . . . . 6 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F Fn X )
9	5	adantr 453	. . . . . . . . . . 11 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> F : X --> ( 0 [,] +oo ) )
10		simprr 735	. . . . . . . . . . 11 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> w e. X )
11	9, 10	ffvelrnd 5874	. . . . . . . . . 10 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) e. ( 0 [,] +oo ) )
12		elxrge0 11013	. . . . . . . . . . 11 |- ( ( F ` w ) e. ( 0 [,] +oo ) <-> ( ( F ` w ) e. RR* /\ 0 <_ ( F ` w ) ) )
13	12	simplbi 448	. . . . . . . . . 10 |- ( ( F ` w ) e. ( 0 [,] +oo ) -> ( F ` w ) e. RR* )
14	11, 13	syl 16	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) e. RR* )
15		simpll 732	. . . . . . . . . 10 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> D e. ( Met ` X ) )
16		simpr 449	. . . . . . . . . . . 12 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> S C_ X )
17	16	sselda 3350	. . . . . . . . . . 11 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> z e. X )
18	17	adantrr 699	. . . . . . . . . 10 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> z e. X )
19		metcl 18367	. . . . . . . . . 10 |- ( ( D e. ( Met ` X ) /\ z e. X /\ w e. X ) -> ( z D w ) e. RR )
20	15, 18, 10, 19	syl3anc 1185	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( z D w ) e. RR )
21	12	simprbi 452	. . . . . . . . . 10 |- ( ( F ` w ) e. ( 0 [,] +oo ) -> 0 <_ ( F ` w ) )
22	11, 21	syl 16	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> 0 <_ ( F ` w ) )
23	3	metdsle 18887	. . . . . . . . . 10 |- ( ( ( D e. ( * Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) <_ ( z D w ) )
24	2, 23	sylanl1 633	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) <_ ( z D w ) )
25		xrrege0 10767	. . . . . . . . 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 1186	. . . . . . . 8 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) e. RR )
27	26	anassrs 631	. . . . . . 7 |- ( ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) /\ w e. X ) -> ( F ` w ) e. RR )
28	27	ralrimiva 2791	. . . . . 6 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> A. w e. X ( F ` w ) e. RR )
29		ffnfv 5897	. . . . . 6 |- ( F : X --> RR <-> ( F Fn X /\ A. w e. X ( F ` w ) e. RR ) )
30	8, 28, 29	sylanbrc 647	. . . . 5 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F : X --> RR )
31	30	ex 425	. . . 4 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( z e. S -> F : X --> RR ) )
32	31	exlimdv 1647	. . 3 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( E. z z e. S -> F : X --> RR ) )
33	1, 32	syl5bi 210	. 2 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( S =/= (/) -> F : X --> RR ) )
34	33	3impia 1151	1 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F : X --> RR )

Syntax hints:   -> wi 4   /\ wa 360   /\ w3a 937   E.wex 1551   = wceq 1653   e. wcel 1726   =/= wne 2601   A.wral 2707   C_ wss 3322   (/)c0 3630   class class class wbr 4215   e. cmpt 4269   `'ccnv 4880   ran crn 4882   Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   supcsup 7448   RRcr 8994   0cc0 8995   +oocpnf 9122   RR*cxr 9124   < clt 9125   <_ cle 9126   [,]cicc 10924   * Metcxmt 16691   Metcme 16692

This theorem is referenced by:  metdscn2  18892  lebnumlem1  18991  lebnumlem3  18993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-er 6908  df-ec 6910  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-2 10063  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-icc 10928  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702