Theorem metds0 22653

Description: If a point is in a set, its distance to the set is zero. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)

Hypothesis

Ref	Expression
metdscn.f	|- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )

Assertion

Ref	Expression
metds0	|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( F ` A ) = 0 )

Distinct variable groups:   x, y, A   x, D, y   x, S, y   x, X, y

Allowed substitution hints:   F( x, y)

Proof of Theorem metds0

Step	Hyp	Ref	Expression
1		metdscn.f	. . . . . . . . . 10 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
2	1	metdsf 22651	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
3	2	3adant3 1081	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> F : X --> ( 0 [,] +oo ) )
4		ssel2 3598	. . . . . . . . 9 |- ( ( S C_ X /\ A e. S ) -> A e. X )
5	4	3adant1 1079	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> A e. X )
6	3, 5	ffvelrnd 6360	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( F ` A ) e. ( 0 [,] +oo ) )
7		elxrge0 12281	. . . . . . . 8 |- ( ( F ` A ) e. ( 0 [,] +oo ) <-> ( ( F ` A ) e. RR* /\ 0 <_ ( F ` A ) ) )
8	7	simplbi 476	. . . . . . 7 |- ( ( F ` A ) e. ( 0 [,] +oo ) -> ( F ` A ) e. RR* )
9	6, 8	syl 17	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( F ` A ) e. RR* )
10		xrleid 11983	. . . . . 6 |- ( ( F ` A ) e. RR* -> ( F ` A ) <_ ( F ` A ) )
11	9, 10	syl 17	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( F ` A ) <_ ( F ` A ) )
12		simp1 1061	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> D e. ( *Met ` X ) )
13		simp2 1062	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> S C_ X )
14	1	metdsge 22652	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) e. RR* ) -> ( ( F ` A ) <_ ( F ` A ) <-> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) = (/) ) )
15	12, 13, 5, 9, 14	syl31anc 1329	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( ( F ` A ) <_ ( F ` A ) <-> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) = (/) ) )
16	11, 15	mpbid 222	. . . 4 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) = (/) )
17		simpl3 1066	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) /\ 0 < ( F ` A ) ) -> A e. S )
18	12	adantr 481	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) /\ 0 < ( F ` A ) ) -> D e. ( *Met ` X ) )
19	5	adantr 481	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) /\ 0 < ( F ` A ) ) -> A e. X )
20	9	adantr 481	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) /\ 0 < ( F ` A ) ) -> ( F ` A ) e. RR* )
21		simpr 477	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) /\ 0 < ( F ` A ) ) -> 0 < ( F ` A ) )
22		xblcntr 22216	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ ( ( F ` A ) e. RR* /\ 0 < ( F ` A ) ) ) -> A e. ( A ( ball ` D ) ( F ` A ) ) )
23	18, 19, 20, 21, 22	syl112anc 1330	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) /\ 0 < ( F ` A ) ) -> A e. ( A ( ball ` D ) ( F ` A ) ) )
24		inelcm 4032	. . . . . . 7 |- ( ( A e. S /\ A e. ( A ( ball ` D ) ( F ` A ) ) ) -> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) =/= (/) )
25	17, 23, 24	syl2anc 693	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) /\ 0 < ( F ` A ) ) -> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) =/= (/) )
26	25	ex 450	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( 0 < ( F ` A ) -> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) =/= (/) ) )
27	26	necon2bd 2810	. . . 4 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) = (/) -> -. 0 < ( F ` A ) ) )
28	16, 27	mpd 15	. . 3 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> -. 0 < ( F ` A ) )
29	7	simprbi 480	. . . . . 6 |- ( ( F ` A ) e. ( 0 [,] +oo ) -> 0 <_ ( F ` A ) )
30	6, 29	syl 17	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> 0 <_ ( F ` A ) )
31		0xr 10086	. . . . . 6 |- 0 e. RR*
32		xrleloe 11977	. . . . . 6 |- ( ( 0 e. RR* /\ ( F ` A ) e. RR* ) -> ( 0 <_ ( F ` A ) <-> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) ) )
33	31, 9, 32	sylancr 695	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( 0 <_ ( F ` A ) <-> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) ) )
34	30, 33	mpbid 222	. . . 4 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) )
35	34	ord 392	. . 3 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( -. 0 < ( F ` A ) -> 0 = ( F ` A ) ) )
36	28, 35	mpd 15	. 2 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> 0 = ( F ` A ) )
37	36	eqcomd 2628	1 |- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. S ) -> ( F ` A ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,]cicc 12178   *Metcxmt 19731   ballcbl 19733

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-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-bl 19741

This theorem is referenced by:  metdsle  22655  metnrmlem1  22662