Theorem metustfbas 20146

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 20132	. . . . . 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 5194	. . . . . . . . . 10 |- ( `' D " ( 0 [,) a ) ) C_ dom D
5		psmetf 19887	. . . . . . . . . . . 12 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
6		fdm 5568	. . . . . . . . . . . 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 3409	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> ( `' D " ( 0 [,) a ) ) C_ ( X X. X ) )
10	3, 9	eqsstrd 3395	. . . . . . . 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 2849	. . . . . 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 2803	. . . 4 |- ( D e. (PsMet ` X ) -> A. x e. F x C_ ( X X. X ) )
15		pwssb 4262	. . . 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 6529	. . . . . . 7 |- ( D e. (PsMet ` X ) -> `' D e. _V )
19		imaexg 6520	. . . . . . 7 |- ( `' D e. _V -> ( `' D " ( 0 [,) 1 ) ) e. _V )
20		elisset 2988	. . . . . . 7 |- ( ( `' D " ( 0 [,) 1 ) ) e. _V -> E. x x = ( `' D " ( 0 [,) 1 ) ) )
21		1rp 11000	. . . . . . . . 9 |- 1 e. RR+
22		oveq2 6104	. . . . . . . . . . . 12 |- ( a = 1 -> ( 0 [,) a ) = ( 0 [,) 1 ) )
23	22	imaeq2d 5174	. . . . . . . . . . 11 |- ( a = 1 -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) 1 ) ) )
24	23	eqeq2d 2454	. . . . . . . . . 10 |- ( a = 1 -> ( x = ( `' D " ( 0 [,) a ) ) <-> x = ( `' D " ( 0 [,) 1 ) ) ) )
25	24	rspcev 3078	. . . . . . . . 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 1625	. . . . . . 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 1680	. . . . . 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 3651	. . . 4 |- ( F =/= (/) <-> E. x x e. F )
33	31, 32	sylibr 212	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> F =/= (/) )
34	1	metustid 20140	. . . . . . 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 3651	. . . . . . . . . 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 5127	. . . . . . . . . . 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 3648	. . . . . . . . . 10 |- ( <. p , p >. e. ( _I |` X ) -> ( _I |` X ) =/= (/) )
42	40, 41	syl 16	. . . . . . . . 9 |- ( p e. X -> ( _I |` X ) =/= (/) )
43	42	exlimiv 1688	. . . . . . . 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 3675	. . . . . 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 3173	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> -. (/) e. F )
49		df-nel 2614	. . . 4 |- ( (/) e/ F <-> -. (/) e. F )
50	48, 49	sylibr 212	. . 3 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (/) e/ F )
51		df-ss 3347	. . . . . . . . 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 759	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> x e. F )
55	53, 54	eqeltrd 2517	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) e. F )
56		dfss1 3560	. . . . . . . . 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 760	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> y e. F )
60	58, 59	eqeltrd 2517	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) e. F )
61		simplr 754	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> D e. (PsMet ` X ) )
62		simprl 755	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> x e. F )
63		simprr 756	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> y e. F )
64	1	metustto 20138	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. F /\ y e. F ) -> ( x C_ y \/ y C_ x ) )
65	61, 62, 63, 64	syl3anc 1218	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x C_ y \/ y C_ x ) )
66	55, 60, 65	mpjaodan 784	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) e. F )
67		vex 2980	. . . . . . . . 9 |- x e. _V
68	67	inex1 4438	. . . . . . . 8 |- ( x i^i y ) e. _V
69	68	pwid 3879	. . . . . . 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 3875	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) C_ ( x i^i y ) )
72		sseq1 3382	. . . . . 6 |- ( z = ( x i^i y ) -> ( z C_ ( x i^i y ) <-> ( x i^i y ) C_ ( x i^i y ) ) )
73	72	rspcev 3078	. . . . 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 2813	. . 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 1168	. 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 5722	. . . . 5 |- ( D e. (PsMet ` X ) -> X e. _V )
78	77	adantl 466	. . . 4 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> X e. _V )
79		xpexg 6512	. . . 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 19413	. . 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 913	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 965   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2611   e/ wnel 2612   A.wral 2720   E.wrex 2721   _Vcvv 2977   i^i cin 3332   C_ wss 3333   (/)c0 3642   ~Pcpw 3865   <.cop 3888   e. cmpt 4355   _I cid 4636   X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846   |` cres 4847   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096   0cc0 9287   1c1 9288   RR*cxr 9422   RR+crp 10996   [,)cico 11307  PsMetcpsmet 17805   fBascfbas 17809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-rp 10997  df-ico 11311  df-psmet 17814  df-fbas 17819

This theorem is referenced by:  metust  20148  cfilucfil  20150  metuel  20158  psmetutop  20163  restmetu  20167  metucn  20169