Theorem metdscn2 22660

Description: The function F which gives the distance from a point to a nonempty set in a metric space is a continuous function into the topology of the complex numbers. (Contributed by Mario Carneiro, 5-Sep-2015.)

Hypotheses

Ref	Expression
metdscn.f	|- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
metdscn.j	|- J = ( MetOpen ` D )
metdscn2.k	|- K = ( TopOpen ` ℂfld )

Assertion

Ref	Expression
metdscn2	|- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F e. ( J Cn K ) )

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

Allowed substitution hints:   F( x, y)   J( x)   K( x, y)

Proof of Theorem metdscn2

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7 |- ( dist ` RR*s ) = ( dist ` RR*s )
2	1	xrsdsre 22613	. . . . . 6 |- ( ( dist ` RR*s ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
3	1	xrsxmet 22612	. . . . . . 7 |- ( dist ` RR*s ) e. ( *Met ` RR* )
4		ressxr 10083	. . . . . . 7 |- RR C_ RR*
5		eqid 2622	. . . . . . . 8 |- ( ( dist ` RR*s ) |` ( RR X. RR ) ) = ( ( dist ` RR*s ) |` ( RR X. RR ) )
6		eqid 2622	. . . . . . . 8 |- ( MetOpen ` ( dist ` RR*s ) ) = ( MetOpen ` ( dist ` RR*s ) )
7		eqid 2622	. . . . . . . 8 |- ( MetOpen ` ( ( dist ` RR*s ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( dist ` RR*s ) |` ( RR X. RR ) ) )
8	5, 6, 7	metrest 22329	. . . . . . 7 |- ( ( ( dist ` RR*s ) e. ( *Met ` RR* ) /\ RR C_ RR* ) -> ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) = ( MetOpen ` ( ( dist ` RR*s ) |` ( RR X. RR ) ) ) )
9	3, 4, 8	mp2an 708	. . . . . 6 |- ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) = ( MetOpen ` ( ( dist ` RR*s ) |` ( RR X. RR ) ) )
10	2, 9	tgioo 22599	. . . . 5 |- ( topGen ` ran (,) ) = ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR )
11		metdscn2.k	. . . . . 6 |- K = ( TopOpen ` ℂfld )
12	11	tgioo2 22606	. . . . 5 |- ( topGen ` ran (,) ) = ( K ↾t RR )
13	10, 12	eqtr3i 2646	. . . 4 |- ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) = ( K ↾t RR )
14	13	oveq2i 6661	. . 3 |- ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) ) = ( J Cn ( K ↾t RR ) )
15	11	cnfldtop 22587	. . . 4 |- K e. Top
16		cnrest2r 21091	. . . 4 |- ( K e. Top -> ( J Cn ( K ↾t RR ) ) C_ ( J Cn K ) )
17	15, 16	ax-mp 5	. . 3 |- ( J Cn ( K ↾t RR ) ) C_ ( J Cn K )
18	14, 17	eqsstri 3635	. 2 |- ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) ) C_ ( J Cn K )
19		metxmet 22139	. . . . 5 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
20		metdscn.f	. . . . . 6 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
21		metdscn.j	. . . . . 6 |- J = ( MetOpen ` D )
22	20, 21, 1, 6	metdscn 22659	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) )
23	19, 22	sylan 488	. . . 4 |- ( ( D e. ( Met ` X ) /\ S C_ X ) -> F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) )
24	23	3adant3 1081	. . 3 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) )
25	20	metdsre 22656	. . . 4 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F : X --> RR )
26		frn 6053	. . . 4 |- ( F : X --> RR -> ran F C_ RR )
27	6	mopntopon 22244	. . . . . 6 |- ( ( dist ` RR*s ) e. ( *Met ` RR* ) -> ( MetOpen ` ( dist ` RR*s ) ) e. (TopOn ` RR* ) )
28	3, 27	ax-mp 5	. . . . 5 |- ( MetOpen ` ( dist ` RR*s ) ) e. (TopOn ` RR* )
29		cnrest2 21090	. . . . 5 |- ( ( ( MetOpen ` ( dist ` RR*s ) ) e. (TopOn ` RR* ) /\ ran F C_ RR /\ RR C_ RR* ) -> ( F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) <-> F e. ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) ) ) )
30	28, 4, 29	mp3an13 1415	. . . 4 |- ( ran F C_ RR -> ( F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) <-> F e. ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) ) ) )
31	25, 26, 30	3syl 18	. . 3 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> ( F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) <-> F e. ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) ) ) )
32	24, 31	mpbid 222	. 2 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F e. ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) ↾t RR ) ) )
33	18, 32	sseldi 3601	1 |- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   X. cxp 5112   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   RR*cxr 10073   < clt 10074   (,)cioo 12175   distcds 15950   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098   RR*scxrs 16160   *Metcxmt 19731   Metcme 19732   MetOpencmopn 19736  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   Cn ccn 21028

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-rep 4771  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-xrs 16162  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-xms 22125  df-ms 22126

This theorem is referenced by:  lebnumlem2  22761