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.

Step	Hyp	Ref	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* , `' < ) )
4	3	metdsf 18839	. . . . . . . . 9 |- ( ( D e. ( * Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
5	2, 4	sylan 458	. . . . . . . 8 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
6	5	adantr 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 )
8	6, 7	syl 16	. . . . . 6 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F Fn X )
9	5	adantr 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 )
11	9, 10	ffvelrnd 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 ) ) )
13	12	simplbi 447	. . . . . . . . . 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 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 )
17	16	sselda 3316	. . . . . . . . . . 11 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> z e. X )
18	17	adantrr 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 )
20	15, 18, 10, 19	syl3anc 1184	. . . . . . . . 9 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( z D w ) e. RR )
21	12	simprbi 451	. . . . . . . . . 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 18843	. . . . . . . . . 10 |- ( ( ( D e. ( * Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) <_ ( z D w ) )
24	2, 23	sylanl1 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 )
26	14, 20, 22, 24, 25	syl22anc 1185	. . . . . . . 8 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ ( z e. S /\ w e. X ) ) -> ( F ` w ) e. RR )
27	26	anassrs 630	. . . . . . 7 |- ( ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) /\ w e. X ) -> ( F ` w ) e. RR )
28	27	ralrimiva 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 ) )
30	8, 28, 29	sylanbrc 646	. . . . 5 |- ( ( ( D e. ( Met ` X ) /\ S C_ X ) /\ z e. S ) -> F : X --> RR )
31	30	ex 424	. . . 4 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( z e. S -> F : X --> RR ) )
32	31	exlimdv 1643	. . 3 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( E. z z e. S -> F : X --> RR ) )
33	1, 32	syl5bi 209	. 2 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> ( S =/= (/) -> F : X --> RR ) )
34	33	3impia 1150	1 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F : X --> RR )

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