Theorem metustfbas 22362

Description: The filter base generated by a metric D. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
metust.1	|- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )

Assertion

Ref	Expression
metustfbas	|- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F e. ( fBas ` ( X X. X ) ) )

Distinct variable groups:   D, a   X, a   F, a

Proof of Theorem metustfbas

Dummy variables p x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metust.1	. . . . . . 7 |- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
2	1	metustel 22355	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( x e. F <-> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) )
3		simpr 477	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x = ( `' D " ( 0 [,) a ) ) )
4		cnvimass 5485	. . . . . . . . . 10 |- ( `' D " ( 0 [,) a ) ) C_ dom D
5		psmetf 22111	. . . . . . . . . . . 12 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
6		fdm 6051	. . . . . . . . . . . 12 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
7	5, 6	syl 17	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> dom D = ( X X. X ) )
8	7	adantr 481	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> dom D = ( X X. X ) )
9	4, 8	syl5sseq 3653	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> ( `' D " ( 0 [,) a ) ) C_ ( X X. X ) )
10	3, 9	eqsstrd 3639	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x C_ ( X X. X ) )
11	10	ex 450	. . . . . . 7 |- ( D e. (PsMet ` X ) -> ( x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) )
12	11	rexlimdvw 3034	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) )
13	2, 12	sylbid 230	. . . . 5 |- ( D e. (PsMet ` X ) -> ( x e. F -> x C_ ( X X. X ) ) )
14	13	ralrimiv 2965	. . . 4 |- ( D e. (PsMet ` X ) -> A. x e. F x C_ ( X X. X ) )
15		pwssb 4612	. . . 4 |- ( F C_ ~P ( X X. X ) <-> A. x e. F x C_ ( X X. X ) )
16	14, 15	sylibr 224	. . 3 |- ( D e. (PsMet ` X ) -> F C_ ~P ( X X. X ) )
17	16	adantl 482	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F C_ ~P ( X X. X ) )
18		cnvexg 7112	. . . . . . 7 |- ( D e. (PsMet ` X ) -> `' D e. _V )
19		imaexg 7103	. . . . . . 7 |- ( `' D e. _V -> ( `' D " ( 0 [,) 1 ) ) e. _V )
20		elisset 3215	. . . . . . 7 |- ( ( `' D " ( 0 [,) 1 ) ) e. _V -> E. x x = ( `' D " ( 0 [,) 1 ) ) )
21		1rp 11836	. . . . . . . . 9 |- 1 e. RR+
22		oveq2 6658	. . . . . . . . . . . 12 |- ( a = 1 -> ( 0 [,) a ) = ( 0 [,) 1 ) )
23	22	imaeq2d 5466	. . . . . . . . . . 11 |- ( a = 1 -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) 1 ) ) )
24	23	eqeq2d 2632	. . . . . . . . . 10 |- ( a = 1 -> ( x = ( `' D " ( 0 [,) a ) ) <-> x = ( `' D " ( 0 [,) 1 ) ) ) )
25	24	rspcev 3309	. . . . . . . . 9 |- ( ( 1 e. RR+ /\ x = ( `' D " ( 0 [,) 1 ) ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
26	21, 25	mpan 706	. . . . . . . 8 |- ( x = ( `' D " ( 0 [,) 1 ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
27	26	eximi 1762	. . . . . . 7 |- ( E. x x = ( `' D " ( 0 [,) 1 ) ) -> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
28	18, 19, 20, 27	4syl 19	. . . . . 6 |- ( D e. (PsMet ` X ) -> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
29	2	exbidv 1850	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( E. x x e. F <-> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) )
30	28, 29	mpbird 247	. . . . 5 |- ( D e. (PsMet ` X ) -> E. x x e. F )
31	30	adantl 482	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> E. x x e. F )
32		n0 3931	. . . 4 |- ( F =/= (/) <-> E. x x e. F )
33	31, 32	sylibr 224	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F =/= (/) )
34	1	metustid 22359	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F ) -> ( _I |` X ) C_ x )
35	34	adantll 750	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) C_ x )
36		n0 3931	. . . . . . . . . 10 |- ( X =/= (/) <-> E. p p e. X )
37	36	biimpi 206	. . . . . . . . 9 |- ( X =/= (/) -> E. p p e. X )
38	37	adantr 481	. . . . . . . 8 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> E. p p e. X )
39		opelresi 5408	. . . . . . . . . . 11 |- ( p e. X -> ( <. p , p >. e. ( _I |` X ) <-> p e. X ) )
40	39	ibir 257	. . . . . . . . . 10 |- ( p e. X -> <. p , p >. e. ( _I |` X ) )
41		ne0i 3921	. . . . . . . . . 10 |- ( <. p , p >. e. ( _I |` X ) -> ( _I |` X ) =/= (/) )
42	40, 41	syl 17	. . . . . . . . 9 |- ( p e. X -> ( _I |` X ) =/= (/) )
43	42	exlimiv 1858	. . . . . . . 8 |- ( E. p p e. X -> ( _I |` X ) =/= (/) )
44	38, 43	syl 17	. . . . . . 7 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( _I |` X ) =/= (/) )
45	44	adantr 481	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) =/= (/) )
46		ssn0 3976	. . . . . 6 |- ( ( ( _I |` X ) C_ x /\ ( _I |` X ) =/= (/) ) -> x =/= (/) )
47	35, 45, 46	syl2anc 693	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> x =/= (/) )
48	47	nelrdva 3417	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> -. (/) e. F )
49		df-nel 2898	. . . 4 |- ( (/) e/ F <-> -. (/) e. F )
50	48, 49	sylibr 224	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (/) e/ F )
51		df-ss 3588	. . . . . . . . 9 |- ( x C_ y <-> ( x i^i y ) = x )
52	51	biimpi 206	. . . . . . . 8 |- ( x C_ y -> ( x i^i y ) = x )
53	52	adantl 482	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) = x )
54		simplrl 800	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> x e. F )
55	53, 54	eqeltrd 2701	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) e. F )
56		sseqin2 3817	. . . . . . . . 9 |- ( y C_ x <-> ( x i^i y ) = y )
57	56	biimpi 206	. . . . . . . 8 |- ( y C_ x -> ( x i^i y ) = y )
58	57	adantl 482	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) = y )
59		simplrr 801	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> y e. F )
60	58, 59	eqeltrd 2701	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) e. F )
61		simplr 792	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> D e. (PsMet ` X ) )
62		simprl 794	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> x e. F )
63		simprr 796	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> y e. F )
64	1	metustto 22358	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F /\ y e. F ) -> ( x C_ y \/ y C_ x ) )
65	61, 62, 63, 64	syl3anc 1326	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x C_ y \/ y C_ x ) )
66	55, 60, 65	mpjaodan 827	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) e. F )
67		vex 3203	. . . . . . . . 9 |- x e. _V
68	67	inex1 4799	. . . . . . . 8 |- ( x i^i y ) e. _V
69	68	pwid 4174	. . . . . . 7 |- ( x i^i y ) e. ~P ( x i^i y )
70	69	a1i 11	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) e. ~P ( x i^i y ) )
71	70	elpwid 4170	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) C_ ( x i^i y ) )
72		sseq1 3626	. . . . . 6 |- ( z = ( x i^i y ) -> ( z C_ ( x i^i y ) <-> ( x i^i y ) C_ ( x i^i y ) ) )
73	72	rspcev 3309	. . . . 5 |- ( ( ( x i^i y ) e. F /\ ( x i^i y ) C_ ( x i^i y ) ) -> E. z e. F z C_ ( x i^i y ) )
74	66, 71, 73	syl2anc 693	. . . 4 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> E. z e. F z C_ ( x i^i y ) )
75	74	ralrimivva 2971	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) )
76	33, 50, 75	3jca 1242	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) ) )
77		elfvex 6221	. . . . 5 |- ( D e. (PsMet ` X ) -> X e. _V )
78	77	adantl 482	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> X e. _V )
79		xpexg 6960	. . . 4 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
80	78, 78, 79	syl2anc 693	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( X X. X ) e. _V )
81		isfbas2 21639	. . 3 |- ( ( X X. X ) e. _V -> ( F e. ( fBas ` ( X X. X ) ) <-> ( F C_ ~P ( X X. X ) /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) ) ) ) )
82	80, 81	syl 17	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( F e. ( fBas ` ( X X. X ) ) <-> ( F C_ ~P ( X X. X ) /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) ) ) ) )
83	17, 76, 82	mpbir2and 957	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F e. ( fBas ` ( X X. X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   RR*cxr 10073   RR+crp 11832   [,)cico 12177  PsMetcpsmet 19730   fBascfbas 19734

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

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-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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-rp 11833  df-ico 12181  df-psmet 19738  df-fbas 19743

This theorem is referenced by:  metust  22363  cfilucfil  22364  metuel  22369  psmetutop  22372  restmetu  22375  metucn  22376