Theorem metustfbas 21194

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 21180	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( x e. F <-> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) )
3		simpr 461	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x = ( `' D " ( 0 [,) a ) ) )
4		cnvimass 5367	. . . . . . . . . 10 |- ( `' D " ( 0 [,) a ) ) C_ dom D
5		psmetf 20935	. . . . . . . . . . . 12 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
6		fdm 5741	. . . . . . . . . . . 12 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
7	5, 6	syl 16	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> dom D = ( X X. X ) )
8	7	adantr 465	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> dom D = ( X X. X ) )
9	4, 8	syl5sseq 3547	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> ( `' D " ( 0 [,) a ) ) C_ ( X X. X ) )
10	3, 9	eqsstrd 3533	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x C_ ( X X. X ) )
11	10	ex 434	. . . . . . 7 |- ( D e. (PsMet ` X ) -> ( x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) )
12	11	rexlimdvw 2952	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) )
13	2, 12	sylbid 215	. . . . 5 |- ( D e. (PsMet ` X ) -> ( x e. F -> x C_ ( X X. X ) ) )
14	13	ralrimiv 2869	. . . 4 |- ( D e. (PsMet ` X ) -> A. x e. F x C_ ( X X. X ) )
15		pwssb 4422	. . . 4 |- ( F C_ ~P ( X X. X ) <-> A. x e. F x C_ ( X X. X ) )
16	14, 15	sylibr 212	. . 3 |- ( D e. (PsMet ` X ) -> F C_ ~P ( X X. X ) )
17	16	adantl 466	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F C_ ~P ( X X. X ) )
18		cnvexg 6745	. . . . . . 7 |- ( D e. (PsMet ` X ) -> `' D e. _V )
19		imaexg 6736	. . . . . . 7 |- ( `' D e. _V -> ( `' D " ( 0 [,) 1 ) ) e. _V )
20		elisset 3120	. . . . . . 7 |- ( ( `' D " ( 0 [,) 1 ) ) e. _V -> E. x x = ( `' D " ( 0 [,) 1 ) ) )
21		1rp 11249	. . . . . . . . 9 |- 1 e. RR+
22		oveq2 6304	. . . . . . . . . . . 12 |- ( a = 1 -> ( 0 [,) a ) = ( 0 [,) 1 ) )
23	22	imaeq2d 5347	. . . . . . . . . . 11 |- ( a = 1 -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) 1 ) ) )
24	23	eqeq2d 2471	. . . . . . . . . 10 |- ( a = 1 -> ( x = ( `' D " ( 0 [,) a ) ) <-> x = ( `' D " ( 0 [,) 1 ) ) ) )
25	24	rspcev 3210	. . . . . . . . 9 |- ( ( 1 e. RR+ /\ x = ( `' D " ( 0 [,) 1 ) ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
26	21, 25	mpan 670	. . . . . . . 8 |- ( x = ( `' D " ( 0 [,) 1 ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
27	26	eximi 1657	. . . . . . 7 |- ( E. x x = ( `' D " ( 0 [,) 1 ) ) -> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
28	18, 19, 20, 27	4syl 21	. . . . . 6 |- ( D e. (PsMet ` X ) -> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) )
29	2	exbidv 1715	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( E. x x e. F <-> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) )
30	28, 29	mpbird 232	. . . . 5 |- ( D e. (PsMet ` X ) -> E. x x e. F )
31	30	adantl 466	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> E. x x e. F )
32		n0 3803	. . . 4 |- ( F =/= (/) <-> E. x x e. F )
33	31, 32	sylibr 212	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F =/= (/) )
34	1	metustid 21188	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F ) -> ( _I |` X ) C_ x )
35	34	adantll 713	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) C_ x )
36		n0 3803	. . . . . . . . . 10 |- ( X =/= (/) <-> E. p p e. X )
37	36	biimpi 194	. . . . . . . . 9 |- ( X =/= (/) -> E. p p e. X )
38	37	adantr 465	. . . . . . . 8 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> E. p p e. X )
39		opelresi 5295	. . . . . . . . . . 11 |- ( p e. X -> ( <. p , p >. e. ( _I |` X ) <-> p e. X ) )
40	39	ibir 242	. . . . . . . . . 10 |- ( p e. X -> <. p , p >. e. ( _I |` X ) )
41		ne0i 3799	. . . . . . . . . 10 |- ( <. p , p >. e. ( _I |` X ) -> ( _I |` X ) =/= (/) )
42	40, 41	syl 16	. . . . . . . . 9 |- ( p e. X -> ( _I |` X ) =/= (/) )
43	42	exlimiv 1723	. . . . . . . 8 |- ( E. p p e. X -> ( _I |` X ) =/= (/) )
44	38, 43	syl 16	. . . . . . 7 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( _I |` X ) =/= (/) )
45	44	adantr 465	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) =/= (/) )
46		ssn0 3827	. . . . . 6 |- ( ( ( _I |` X ) C_ x /\ ( _I |` X ) =/= (/) ) -> x =/= (/) )
47	35, 45, 46	syl2anc 661	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ x e. F ) -> x =/= (/) )
48	47	nelrdva 3309	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> -. (/) e. F )
49		df-nel 2655	. . . 4 |- ( (/) e/ F <-> -. (/) e. F )
50	48, 49	sylibr 212	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (/) e/ F )
51		df-ss 3485	. . . . . . . . 9 |- ( x C_ y <-> ( x i^i y ) = x )
52	51	biimpi 194	. . . . . . . 8 |- ( x C_ y -> ( x i^i y ) = x )
53	52	adantl 466	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) = x )
54		simplrl 761	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> x e. F )
55	53, 54	eqeltrd 2545	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) e. F )
56		dfss1 3699	. . . . . . . . 9 |- ( y C_ x <-> ( x i^i y ) = y )
57	56	biimpi 194	. . . . . . . 8 |- ( y C_ x -> ( x i^i y ) = y )
58	57	adantl 466	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) = y )
59		simplrr 762	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> y e. F )
60	58, 59	eqeltrd 2545	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) e. F )
61		simplr 755	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> D e. (PsMet ` X ) )
62		simprl 756	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> x e. F )
63		simprr 757	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> y e. F )
64	1	metustto 21186	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F /\ y e. F ) -> ( x C_ y \/ y C_ x ) )
65	61, 62, 63, 64	syl3anc 1228	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x C_ y \/ y C_ x ) )
66	55, 60, 65	mpjaodan 786	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) e. F )
67		vex 3112	. . . . . . . . 9 |- x e. _V
68	67	inex1 4597	. . . . . . . 8 |- ( x i^i y ) e. _V
69	68	pwid 4029	. . . . . . 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 4025	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) C_ ( x i^i y ) )
72		sseq1 3520	. . . . . 6 |- ( z = ( x i^i y ) -> ( z C_ ( x i^i y ) <-> ( x i^i y ) C_ ( x i^i y ) ) )
73	72	rspcev 3210	. . . . 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 661	. . . 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 2878	. . 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 1176	. 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 5899	. . . . 5 |- ( D e. (PsMet ` X ) -> X e. _V )
78	77	adantl 466	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> X e. _V )
79		xpexg 6601	. . . 4 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
80	78, 78, 79	syl2anc 661	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( X X. X ) e. _V )
81		isfbas2 20461	. . 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 16	. 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 922	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F e. ( fBas ` ( X X. X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1395   E.wex 1613   e. wcel 1819   =/= wne 2652   e/ wnel 2653   A.wral 2807   E.wrex 2808   _Vcvv 3109   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   <.cop 4038   |-> cmpt 4515   _I cid 4799   X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   |` cres 5010   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   RR*cxr 9644   RR+crp 11245   [,)cico 11556  PsMetcpsmet 18528   fBascfbas 18532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-rp 11246  df-ico 11560  df-psmet 18537  df-fbas 18542

This theorem is referenced by:  metust  21196  cfilucfil  21198  metuel  21206  psmetutop  21211  restmetu  21215  metucn  21217