Theorem metdscn 22659

Description: The function F which gives the distance from a point to a set is a continuous function into the metric topology of the extended reals. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-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 )
metdscn.c	|- C = ( dist ` RR*s )
metdscn.k	|- K = ( MetOpen ` C )

Assertion

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

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

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

Proof of Theorem metdscn

Dummy variables w r z s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metdscn.f	. . . 4 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
2	1	metdsf 22651	. . 3 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) )
3		iccssxr 12256	. . 3 |- ( 0 [,] +oo ) C_ RR*
4		fss 6056	. . 3 |- ( ( F : X --> ( 0 [,] +oo ) /\ ( 0 [,] +oo ) C_ RR* ) -> F : X --> RR* )
5	2, 3, 4	sylancl 694	. 2 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> RR* )
6		simprr 796	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) -> r e. RR+ )
7	5	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> F : X --> RR* )
8		simplrl 800	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> z e. X )
9	7, 8	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( F ` z ) e. RR* )
10		simprl 794	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> w e. X )
11	7, 10	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( F ` w ) e. RR* )
12		metdscn.c	. . . . . . . . 9 |- C = ( dist ` RR*s )
13	12	xrsdsval 19790	. . . . . . . 8 |- ( ( ( F ` z ) e. RR* /\ ( F ` w ) e. RR* ) -> ( ( F ` z ) C ( F ` w ) ) = if ( ( F ` z ) <_ ( F ` w ) , ( ( F ` w ) +e -e ( F ` z ) ) , ( ( F ` z ) +e -e ( F ` w ) ) ) )
14	9, 11, 13	syl2anc 693	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( ( F ` z ) C ( F ` w ) ) = if ( ( F ` z ) <_ ( F ` w ) , ( ( F ` w ) +e -e ( F ` z ) ) , ( ( F ` z ) +e -e ( F ` w ) ) ) )
15		metdscn.j	. . . . . . . . 9 |- J = ( MetOpen ` D )
16		metdscn.k	. . . . . . . . 9 |- K = ( MetOpen ` C )
17		simplll 798	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> D e. ( *Met ` X ) )
18		simpllr 799	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> S C_ X )
19		simplrr 801	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> r e. RR+ )
20		xmetsym 22152	. . . . . . . . . . 11 |- ( ( D e. ( *Met ` X ) /\ w e. X /\ z e. X ) -> ( w D z ) = ( z D w ) )
21	17, 10, 8, 20	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( w D z ) = ( z D w ) )
22		simprr 796	. . . . . . . . . 10 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( z D w ) < r )
23	21, 22	eqbrtrd 4675	. . . . . . . . 9 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( w D z ) < r )
24	1, 15, 12, 16, 17, 18, 10, 8, 19, 23	metdscnlem 22658	. . . . . . . 8 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( ( F ` w ) +e -e ( F ` z ) ) < r )
25	1, 15, 12, 16, 17, 18, 8, 10, 19, 22	metdscnlem 22658	. . . . . . . 8 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( ( F ` z ) +e -e ( F ` w ) ) < r )
26		breq1 4656	. . . . . . . . 9 |- ( ( ( F ` w ) +e -e ( F ` z ) ) = if ( ( F ` z ) <_ ( F ` w ) , ( ( F ` w ) +e -e ( F ` z ) ) , ( ( F ` z ) +e -e ( F ` w ) ) ) -> ( ( ( F ` w ) +e -e ( F ` z ) ) < r <-> if ( ( F ` z ) <_ ( F ` w ) , ( ( F ` w ) +e -e ( F ` z ) ) , ( ( F ` z ) +e -e ( F ` w ) ) ) < r ) )
27		breq1 4656	. . . . . . . . 9 |- ( ( ( F ` z ) +e -e ( F ` w ) ) = if ( ( F ` z ) <_ ( F ` w ) , ( ( F ` w ) +e -e ( F ` z ) ) , ( ( F ` z ) +e -e ( F ` w ) ) ) -> ( ( ( F ` z ) +e -e ( F ` w ) ) < r <-> if ( ( F ` z ) <_ ( F ` w ) , ( ( F ` w ) +e -e ( F ` z ) ) , ( ( F ` z ) +e -e ( F ` w ) ) ) < r ) )
28	26, 27	ifboth 4124	. . . . . . . 8 |- ( ( ( ( F ` w ) +e -e ( F ` z ) ) < r /\ ( ( F ` z ) +e -e ( F ` w ) ) < r ) -> if ( ( F ` z ) <_ ( F ` w ) , ( ( F ` w ) +e -e ( F ` z ) ) , ( ( F ` z ) +e -e ( F ` w ) ) ) < r )
29	24, 25, 28	syl2anc 693	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> if ( ( F ` z ) <_ ( F ` w ) , ( ( F ` w ) +e -e ( F ` z ) ) , ( ( F ` z ) +e -e ( F ` w ) ) ) < r )
30	14, 29	eqbrtrd 4675	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ ( w e. X /\ ( z D w ) < r ) ) -> ( ( F ` z ) C ( F ` w ) ) < r )
31	30	expr 643	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) /\ w e. X ) -> ( ( z D w ) < r -> ( ( F ` z ) C ( F ` w ) ) < r ) )
32	31	ralrimiva 2966	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) -> A. w e. X ( ( z D w ) < r -> ( ( F ` z ) C ( F ` w ) ) < r ) )
33		breq2 4657	. . . . . . 7 |- ( s = r -> ( ( z D w ) < s <-> ( z D w ) < r ) )
34	33	imbi1d 331	. . . . . 6 |- ( s = r -> ( ( ( z D w ) < s -> ( ( F ` z ) C ( F ` w ) ) < r ) <-> ( ( z D w ) < r -> ( ( F ` z ) C ( F ` w ) ) < r ) ) )
35	34	ralbidv 2986	. . . . 5 |- ( s = r -> ( A. w e. X ( ( z D w ) < s -> ( ( F ` z ) C ( F ` w ) ) < r ) <-> A. w e. X ( ( z D w ) < r -> ( ( F ` z ) C ( F ` w ) ) < r ) ) )
36	35	rspcev 3309	. . . 4 |- ( ( r e. RR+ /\ A. w e. X ( ( z D w ) < r -> ( ( F ` z ) C ( F ` w ) ) < r ) ) -> E. s e. RR+ A. w e. X ( ( z D w ) < s -> ( ( F ` z ) C ( F ` w ) ) < r ) )
37	6, 32, 36	syl2anc 693	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ ( z e. X /\ r e. RR+ ) ) -> E. s e. RR+ A. w e. X ( ( z D w ) < s -> ( ( F ` z ) C ( F ` w ) ) < r ) )
38	37	ralrimivva 2971	. 2 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> A. z e. X A. r e. RR+ E. s e. RR+ A. w e. X ( ( z D w ) < s -> ( ( F ` z ) C ( F ` w ) ) < r ) )
39		simpl 473	. . 3 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> D e. ( *Met ` X ) )
40	12	xrsxmet 22612	. . 3 |- C e. ( *Met ` RR* )
41	15, 16	metcn 22348	. . 3 |- ( ( D e. ( *Met ` X ) /\ C e. ( *Met ` RR* ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> RR* /\ A. z e. X A. r e. RR+ E. s e. RR+ A. w e. X ( ( z D w ) < s -> ( ( F ` z ) C ( F ` w ) ) < r ) ) ) )
42	39, 40, 41	sylancl 694	. 2 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> ( F e. ( J Cn K ) <-> ( F : X --> RR* /\ A. z e. X A. r e. RR+ E. s e. RR+ A. w e. X ( ( z D w ) < s -> ( ( F ` z ) C ( F ` w ) ) < r ) ) ) )
43	5, 38, 42	mpbir2and 957	1 |- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ifcif 4086   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   RR+crp 11832   -ecxne 11943   +ecxad 11944   [,]cicc 12178   distcds 15950   RR*scxrs 16160   *Metcxmt 19731   MetOpencmopn 19736   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-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-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-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-tset 15960  df-ple 15961  df-ds 15964  df-topgen 16104  df-xrs 16162  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cnp 21032

This theorem is referenced by:  metdscn2  22660