Theorem metustfbas 21576

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 21569	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( x e. F <-> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) )
3		simpr 463	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x = ( `' D " ( 0 [,) a ) ) )
4		cnvimass 5213	. . . . . . . . . 10 |- ( `' D " ( 0 [,) a ) ) C_ dom D
5		psmetf 21326	. . . . . . . . . . . 12 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
6		fdm 5756	. . . . . . . . . . . 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 467	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> dom D = ( X X. X ) )
9	4, 8	syl5sseq 3518	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> ( `' D " ( 0 [,) a ) ) C_ ( X X. X ) )
10	3, 9	eqsstrd 3504	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x C_ ( X X. X ) )
11	10	ex 436	. . . . . . 7 |- ( D e. (PsMet ` X ) -> ( x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) )
12	11	rexlimdvw 2922	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) )
13	2, 12	sylbid 219	. . . . 5 |- ( D e. (PsMet ` X ) -> ( x e. F -> x C_ ( X X. X ) ) )
14	13	ralrimiv 2839	. . . 4 |- ( D e. (PsMet ` X ) -> A. x e. F x C_ ( X X. X ) )
15		pwssb 4395	. . . 4 |- ( F C_ ~P ( X X. X ) <-> A. x e. F x C_ ( X X. X ) )
16	14, 15	sylibr 216	. . 3 |- ( D e. (PsMet ` X ) -> F C_ ~P ( X X. X ) )
17	16	adantl 468	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F C_ ~P ( X X. X ) )
18		cnvexg 6759	. . . . . . 7 |- ( D e. (PsMet ` X ) -> `' D e. _V )
19		imaexg 6750	. . . . . . 7 |- ( `' D e. _V -> ( `' D " ( 0 [,) 1 ) ) e. _V )
20		elisset 3096	. . . . . . 7 |- ( ( `' D " ( 0 [,) 1 ) ) e. _V -> E. x x = ( `' D " ( 0 [,) 1 ) ) )
21		1rp 11319	. . . . . . . . 9 |- 1 e. RR+
22		oveq2 6319	. . . . . . . . . . . 12 |- ( a = 1 -> ( 0 [,) a ) = ( 0 [,) 1 ) )
23	22	imaeq2d 5193	. . . . . . . . . . 11 |- ( a = 1 -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) 1 ) ) )
24	23	eqeq2d 2437	. . . . . . . . . 10 |- ( a = 1 -> ( x = ( `' D " ( 0 [,) a ) ) <-> x = ( `' D " ( 0 [,) 1 ) ) ) )
25	24	rspcev 3188	. . . . . . . . 9 |- ( ( 1 e. RR+ /\ x = ( `' D " ( 0 [,) 1 ) ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
26	21, 25	mpan 675	. . . . . . . 8 |- ( x = ( `' D " ( 0 [,) 1 ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
27	26	eximi 1702	. . . . . . 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 1763	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( E. x x e. F <-> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) )
30	28, 29	mpbird 236	. . . . 5 |- ( D e. (PsMet ` X ) -> E. x x e. F )
31	30	adantl 468	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> E. x x e. F )
32		n0 3777	. . . 4 |- ( F =/= (/) <-> E. x x e. F )
33	31, 32	sylibr 216	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F =/= (/) )
34	1	metustid 21573	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F ) -> ( _I |` X ) C_ x )
35	34	adantll 719	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) C_ x )
36		n0 3777	. . . . . . . . . 10 |- ( X =/= (/) <-> E. p p e. X )
37	36	biimpi 198	. . . . . . . . 9 |- ( X =/= (/) -> E. p p e. X )
38	37	adantr 467	. . . . . . . 8 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> E. p p e. X )
39		opelresi 5141	. . . . . . . . . . 11 |- ( p e. X -> ( <. p , p >. e. ( _I |` X ) <-> p e. X ) )
40	39	ibir 246	. . . . . . . . . 10 |- ( p e. X -> <. p , p >. e. ( _I |` X ) )
41		ne0i 3773	. . . . . . . . . 10 |- ( <. p , p >. e. ( _I |` X ) -> ( _I |` X ) =/= (/) )
42	40, 41	syl 17	. . . . . . . . 9 |- ( p e. X -> ( _I |` X ) =/= (/) )
43	42	exlimiv 1771	. . . . . . . 8 |- ( E. p p e. X -> ( _I |` X ) =/= (/) )
44	38, 43	syl 17	. . . . . . 7 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( _I |` X ) =/= (/) )
45	44	adantr 467	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) =/= (/) )
46		ssn0 3803	. . . . . 6 |- ( ( ( _I |` X ) C_ x /\ ( _I |` X ) =/= (/) ) -> x =/= (/) )
47	35, 45, 46	syl2anc 666	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> x =/= (/) )
48	47	nelrdva 3287	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> -. (/) e. F )
49		df-nel 2622	. . . 4 |- ( (/) e/ F <-> -. (/) e. F )
50	48, 49	sylibr 216	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (/) e/ F )
51		df-ss 3456	. . . . . . . . 9 |- ( x C_ y <-> ( x i^i y ) = x )
52	51	biimpi 198	. . . . . . . 8 |- ( x C_ y -> ( x i^i y ) = x )
53	52	adantl 468	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) = x )
54		simplrl 769	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> x e. F )
55	53, 54	eqeltrd 2512	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) e. F )
56		dfss1 3673	. . . . . . . . 9 |- ( y C_ x <-> ( x i^i y ) = y )
57	56	biimpi 198	. . . . . . . 8 |- ( y C_ x -> ( x i^i y ) = y )
58	57	adantl 468	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) = y )
59		simplrr 770	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> y e. F )
60	58, 59	eqeltrd 2512	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) e. F )
61		simplr 761	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> D e. (PsMet ` X ) )
62		simprl 763	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> x e. F )
63		simprr 765	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> y e. F )
64	1	metustto 21572	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F /\ y e. F ) -> ( x C_ y \/ y C_ x ) )
65	61, 62, 63, 64	syl3anc 1265	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x C_ y \/ y C_ x ) )
66	55, 60, 65	mpjaodan 794	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) e. F )
67		vex 3088	. . . . . . . . 9 |- x e. _V
68	67	inex1 4571	. . . . . . . 8 |- ( x i^i y ) e. _V
69	68	pwid 4001	. . . . . . 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 3997	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) C_ ( x i^i y ) )
72		sseq1 3491	. . . . . 6 |- ( z = ( x i^i y ) -> ( z C_ ( x i^i y ) <-> ( x i^i y ) C_ ( x i^i y ) ) )
73	72	rspcev 3188	. . . . 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 666	. . . 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 2848	. . 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 1186	. 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 5914	. . . . 5 |- ( D e. (PsMet ` X ) -> X e. _V )
78	77	adantl 468	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> X e. _V )
79		xpexg 6613	. . . 4 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
80	78, 78, 79	syl2anc 666	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( X X. X ) e. _V )
81		isfbas2 20854	. . 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 931	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F e. ( fBas ` ( X X. X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 983   = wceq 1438   E.wex 1658   e. wcel 1873   =/= wne 2619   e/ wnel 2620   A.wral 2776   E.wrex 2777   _Vcvv 3085   i^i cin 3441   C_ wss 3442   (/)c0 3767   ~Pcpw 3987   <.cop 4010   |-> cmpt 4488   _I cid 4769   X. cxp 4857   `'ccnv 4858   dom cdm 4859   ran crn 4860   |` cres 4861   "cima 4862   -->wf 5603   ` cfv 5607  (class class class)co 6311   0cc0 9552   1c1 9553   RR*cxr 9687   RR+crp 11315   [,)cico 11650  PsMetcpsmet 18959   fBascfbas 18963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-cnex 9608  ax-resscn 9609  ax-1cn 9610  ax-icn 9611  ax-addcl 9612  ax-addrcl 9613  ax-mulcl 9614  ax-mulrcl 9615  ax-mulcom 9616  ax-addass 9617  ax-mulass 9618  ax-distr 9619  ax-i2m1 9620  ax-1ne0 9621  ax-1rid 9622  ax-rnegex 9623  ax-rrecex 9624  ax-cnre 9625  ax-pre-lttri 9626  ax-pre-lttrn 9627  ax-pre-ltadd 9628  ax-pre-mulgt0 9629

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-op 4011  df-uni 4226  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-id 4774  df-po 4780  df-so 4781  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-riota 6273  df-ov 6314  df-oprab 6315  df-mpt2 6316  df-1st 6813  df-2nd 6814  df-er 7380  df-map 7491  df-en 7587  df-dom 7588  df-sdom 7589  df-pnf 9690  df-mnf 9691  df-xr 9692  df-ltxr 9693  df-le 9694  df-sub 9875  df-neg 9876  df-rp 11316  df-ico 11654  df-psmet 18967  df-fbas 18972

This theorem is referenced by:  metust  21577  cfilucfil  21578  metuel  21583  psmetutop  21586  restmetu  21589  metucn  21590