Theorem metdsf 22651

Description: The distance from a point to a set is a nonnegative extended real number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)

Hypothesis

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

Assertion

Ref	Expression
metdsf	|- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )

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

Allowed substitution hints:   F( x, y)

Proof of Theorem metdsf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplll 798	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) /\ y e. S ) -> D e. ( *Met ` X ) )
2		simplr 792	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) /\ y e. S ) -> x e. X )
3		simplr 792	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> S C_ X )
4	3	sselda 3603	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) /\ y e. S ) -> y e. X )
5		xmetcl 22136	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) e. RR* )
6	1, 2, 4, 5	syl3anc 1326	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) /\ y e. S ) -> ( x D y ) e. RR* )
7		eqid 2622	. . . . . 6 |- ( y e. S |-> ( x D y ) ) = ( y e. S |-> ( x D y ) )
8	6, 7	fmptd 6385	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> ( y e. S |-> ( x D y ) ) : S --> RR* )
9		frn 6053	. . . . 5 |- ( ( y e. S |-> ( x D y ) ) : S --> RR* -> ran ( y e. S |-> ( x D y ) ) C_ RR* )
10	8, 9	syl 17	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> ran ( y e. S |-> ( x D y ) ) C_ RR* )
11		infxrcl 12163	. . . 4 |- ( ran ( y e. S |-> ( x D y ) ) C_ RR* -> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) e. RR* )
12	10, 11	syl 17	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) e. RR* )
13		xmetge0 22149	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> 0 <_ ( x D y ) )
14	1, 2, 4, 13	syl3anc 1326	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) /\ y e. S ) -> 0 <_ ( x D y ) )
15	14	ralrimiva 2966	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> A. y e. S 0 <_ ( x D y ) )
16		ovex 6678	. . . . . . 7 |- ( x D y ) e. _V
17	16	rgenw 2924	. . . . . 6 |- A. y e. S ( x D y ) e. _V
18		breq2 4657	. . . . . . 7 |- ( z = ( x D y ) -> ( 0 <_ z <-> 0 <_ ( x D y ) ) )
19	7, 18	ralrnmpt 6368	. . . . . 6 |- ( A. y e. S ( x D y ) e. _V -> ( A. z e. ran ( y e. S |-> ( x D y ) ) 0 <_ z <-> A. y e. S 0 <_ ( x D y ) ) )
20	17, 19	ax-mp 5	. . . . 5 |- ( A. z e. ran ( y e. S |-> ( x D y ) ) 0 <_ z <-> A. y e. S 0 <_ ( x D y ) )
21	15, 20	sylibr 224	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> A. z e. ran ( y e. S |-> ( x D y ) ) 0 <_ z )
22		0xr 10086	. . . . 5 |- 0 e. RR*
23		infxrgelb 12165	. . . . 5 |- ( ( ran ( y e. S |-> ( x D y ) ) C_ RR* /\ 0 e. RR* ) -> ( 0 <_ inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) <-> A. z e. ran ( y e. S |-> ( x D y ) ) 0 <_ z ) )
24	10, 22, 23	sylancl 694	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> ( 0 <_ inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) <-> A. z e. ran ( y e. S |-> ( x D y ) ) 0 <_ z ) )
25	21, 24	mpbird 247	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> 0 <_ inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
26		elxrge0 12281	. . 3 |- (inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) e. ( 0 [,] +oo ) <-> (inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) e. RR* /\ 0 <_ inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) ) )
27	12, 25, 26	sylanbrc 698	. 2 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ x e. X ) -> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) e. ( 0 [,] +oo ) )
28		metdscn.f	. 2 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
29	27, 28	fmptd 6385	1 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   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

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-xmet 19739

This theorem is referenced by:  metds0  22653  metdstri  22654  metdsre  22656  metdseq0  22657  metdscnlem  22658  metdscn  22659  metnrmlem1a  22661  metnrmlem1  22662  lebnumlem1  22760