Theorem metustfbas 21621

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 21614	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( x e. F <-> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) )
3		simpr 467	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x = ( `' D " ( 0 [,) a ) ) )
4		cnvimass 5207	. . . . . . . . . 10 |- ( `' D " ( 0 [,) a ) ) C_ dom D
5		psmetf 21371	. . . . . . . . . . . 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 471	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> dom D = ( X X. X ) )
9	4, 8	syl5sseq 3492	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> ( `' D " ( 0 [,) a ) ) C_ ( X X. X ) )
10	3, 9	eqsstrd 3478	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x C_ ( X X. X ) )
11	10	ex 440	. . . . . . 7 |- ( D e. (PsMet ` X ) -> ( x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) )
12	11	rexlimdvw 2894	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) )
13	2, 12	sylbid 223	. . . . 5 |- ( D e. (PsMet ` X ) -> ( x e. F -> x C_ ( X X. X ) ) )
14	13	ralrimiv 2812	. . . 4 |- ( D e. (PsMet ` X ) -> A. x e. F x C_ ( X X. X ) )
15		pwssb 4382	. . . 4 |- ( F C_ ~P ( X X. X ) <-> A. x e. F x C_ ( X X. X ) )
16	14, 15	sylibr 217	. . 3 |- ( D e. (PsMet ` X ) -> F C_ ~P ( X X. X ) )
17	16	adantl 472	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F C_ ~P ( X X. X ) )
18		cnvexg 6766	. . . . . . 7 |- ( D e. (PsMet ` X ) -> `' D e. _V )
19		imaexg 6757	. . . . . . 7 |- ( `' D e. _V -> ( `' D " ( 0 [,) 1 ) ) e. _V )
20		elisset 3069	. . . . . . 7 |- ( ( `' D " ( 0 [,) 1 ) ) e. _V -> E. x x = ( `' D " ( 0 [,) 1 ) ) )
21		1rp 11335	. . . . . . . . 9 |- 1 e. RR+
22		oveq2 6323	. . . . . . . . . . . 12 |- ( a = 1 -> ( 0 [,) a ) = ( 0 [,) 1 ) )
23	22	imaeq2d 5187	. . . . . . . . . . 11 |- ( a = 1 -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) 1 ) ) )
24	23	eqeq2d 2472	. . . . . . . . . 10 |- ( a = 1 -> ( x = ( `' D " ( 0 [,) a ) ) <-> x = ( `' D " ( 0 [,) 1 ) ) ) )
25	24	rspcev 3162	. . . . . . . . 9 |- ( ( 1 e. RR+ /\ x = ( `' D " ( 0 [,) 1 ) ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
26	21, 25	mpan 681	. . . . . . . 8 |- ( x = ( `' D " ( 0 [,) 1 ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
27	26	eximi 1718	. . . . . . 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 1779	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( E. x x e. F <-> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) )
30	28, 29	mpbird 240	. . . . 5 |- ( D e. (PsMet ` X ) -> E. x x e. F )
31	30	adantl 472	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> E. x x e. F )
32		n0 3753	. . . 4 |- ( F =/= (/) <-> E. x x e. F )
33	31, 32	sylibr 217	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F =/= (/) )
34	1	metustid 21618	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F ) -> ( _I |` X ) C_ x )
35	34	adantll 725	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) C_ x )
36		n0 3753	. . . . . . . . . 10 |- ( X =/= (/) <-> E. p p e. X )
37	36	biimpi 199	. . . . . . . . 9 |- ( X =/= (/) -> E. p p e. X )
38	37	adantr 471	. . . . . . . 8 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> E. p p e. X )
39		opelresi 5135	. . . . . . . . . . 11 |- ( p e. X -> ( <. p , p >. e. ( _I |` X ) <-> p e. X ) )
40	39	ibir 250	. . . . . . . . . 10 |- ( p e. X -> <. p , p >. e. ( _I |` X ) )
41		ne0i 3749	. . . . . . . . . 10 |- ( <. p , p >. e. ( _I |` X ) -> ( _I |` X ) =/= (/) )
42	40, 41	syl 17	. . . . . . . . 9 |- ( p e. X -> ( _I |` X ) =/= (/) )
43	42	exlimiv 1787	. . . . . . . 8 |- ( E. p p e. X -> ( _I |` X ) =/= (/) )
44	38, 43	syl 17	. . . . . . 7 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( _I |` X ) =/= (/) )
45	44	adantr 471	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) =/= (/) )
46		ssn0 3779	. . . . . 6 |- ( ( ( _I |` X ) C_ x /\ ( _I |` X ) =/= (/) ) -> x =/= (/) )
47	35, 45, 46	syl2anc 671	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> x =/= (/) )
48	47	nelrdva 3261	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> -. (/) e. F )
49		df-nel 2636	. . . 4 |- ( (/) e/ F <-> -. (/) e. F )
50	48, 49	sylibr 217	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (/) e/ F )
51		df-ss 3430	. . . . . . . . 9 |- ( x C_ y <-> ( x i^i y ) = x )
52	51	biimpi 199	. . . . . . . 8 |- ( x C_ y -> ( x i^i y ) = x )
53	52	adantl 472	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) = x )
54		simplrl 775	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> x e. F )
55	53, 54	eqeltrd 2540	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) e. F )
56		dfss1 3649	. . . . . . . . 9 |- ( y C_ x <-> ( x i^i y ) = y )
57	56	biimpi 199	. . . . . . . 8 |- ( y C_ x -> ( x i^i y ) = y )
58	57	adantl 472	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) = y )
59		simplrr 776	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> y e. F )
60	58, 59	eqeltrd 2540	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) e. F )
61		simplr 767	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> D e. (PsMet ` X ) )
62		simprl 769	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> x e. F )
63		simprr 771	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> y e. F )
64	1	metustto 21617	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F /\ y e. F ) -> ( x C_ y \/ y C_ x ) )
65	61, 62, 63, 64	syl3anc 1276	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x C_ y \/ y C_ x ) )
66	55, 60, 65	mpjaodan 800	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) e. F )
67		vex 3060	. . . . . . . . 9 |- x e. _V
68	67	inex1 4558	. . . . . . . 8 |- ( x i^i y ) e. _V
69	68	pwid 3977	. . . . . . 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 3973	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) C_ ( x i^i y ) )
72		sseq1 3465	. . . . . 6 |- ( z = ( x i^i y ) -> ( z C_ ( x i^i y ) <-> ( x i^i y ) C_ ( x i^i y ) ) )
73	72	rspcev 3162	. . . . 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 671	. . . 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 2821	. . 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 1194	. 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 5915	. . . . 5 |- ( D e. (PsMet ` X ) -> X e. _V )
78	77	adantl 472	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> X e. _V )
79		xpexg 6620	. . . 4 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
80	78, 78, 79	syl2anc 671	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( X X. X ) e. _V )
81		isfbas2 20899	. . 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 938	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F e. ( fBas ` ( X X. X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   /\ w3a 991   = wceq 1455   E.wex 1674   e. wcel 1898   =/= wne 2633   e/ wnel 2634   A.wral 2749   E.wrex 2750   _Vcvv 3057   i^i cin 3415   C_ wss 3416   (/)c0 3743   ~Pcpw 3963   <.cop 3986   |-> cmpt 4475   _I cid 4763   X. cxp 4851   `'ccnv 4852   dom cdm 4853   ran crn 4854   |` cres 4855   "cima 4856   -->wf 5597   ` cfv 5601  (class class class)co 6315   0cc0 9565   1c1 9566   RR*cxr 9700   RR+crp 11331   [,)cico 11666  PsMetcpsmet 19003   fBascfbas 19007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-rp 11332  df-ico 11670  df-psmet 19011  df-fbas 19016

This theorem is referenced by:  metust  21622  cfilucfil  21623  metuel  21628  psmetutop  21631  restmetu  21634  metucn  21635