MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metustfbas Structured version   Unicode version

Theorem metustfbas 20804
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.
StepHypRef Expression
1 metust.1 . . . . . . 7  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
21metustel 20790 . . . . . 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 5355 . . . . . . . . . 10  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
5 psmetf 20545 . . . . . . . . . . . 12  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
6 fdm 5733 . . . . . . . . . . . 12  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
75, 6syl 16 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
87adantr 465 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  ->  dom  D  =  ( X  X.  X ) )
94, 8syl5sseq 3552 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) a
) )  C_  ( X  X.  X ) )
103, 9eqsstrd 3538 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  ->  x  C_  ( X  X.  X ) )
1110ex 434 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  ( x  =  ( `' D " ( 0 [,) a
) )  ->  x  C_  ( X  X.  X
) ) )
1211rexlimdvw 2958 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( E. a  e.  RR+  x  =  ( `' D "
( 0 [,) a
) )  ->  x  C_  ( X  X.  X
) ) )
132, 12sylbid 215 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  F  ->  x  C_  ( X  X.  X
) ) )
1413ralrimiv 2876 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  F  x  C_  ( X  X.  X ) )
15 pwssb 4412 . . . 4  |-  ( F 
C_  ~P ( X  X.  X )  <->  A. x  e.  F  x  C_  ( X  X.  X ) )
1614, 15sylibr 212 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  F  C_  ~P ( X  X.  X
) )
1716adantl 466 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  F  C_  ~P ( X  X.  X
) )
18 cnvexg 6727 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
19 imaexg 6718 . . . . . . 7  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) 1 ) )  e.  _V )
20 elisset 3124 . . . . . . 7  |-  ( ( `' D " ( 0 [,) 1 ) )  e.  _V  ->  E. x  x  =  ( `' D " ( 0 [,) 1 ) ) )
21 1rp 11220 . . . . . . . . 9  |-  1  e.  RR+
22 oveq2 6290 . . . . . . . . . . . 12  |-  ( a  =  1  ->  (
0 [,) a )  =  ( 0 [,) 1 ) )
2322imaeq2d 5335 . . . . . . . . . . 11  |-  ( a  =  1  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) 1
) ) )
2423eqeq2d 2481 . . . . . . . . . 10  |-  ( a  =  1  ->  (
x  =  ( `' D " ( 0 [,) a ) )  <-> 
x  =  ( `' D " ( 0 [,) 1 ) ) ) )
2524rspcev 3214 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  x  =  ( `' D " ( 0 [,) 1
) ) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
2621, 25mpan 670 . . . . . . . 8  |-  ( x  =  ( `' D " ( 0 [,) 1
) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2726eximi 1635 . . . . . . 7  |-  ( E. x  x  =  ( `' D " ( 0 [,) 1 ) )  ->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2818, 19, 20, 274syl 21 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
292exbidv 1690 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( E. x  x  e.  F  <->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3028, 29mpbird 232 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  E. x  x  e.  F )
3130adantl 466 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  E. x  x  e.  F )
32 n0 3794 . . . 4  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
3331, 32sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  F  =/=  (/) )
341metustid 20798 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  F )  ->  (  _I  |`  X )  C_  x )
3534adantll 713 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  (  _I  |`  X )  C_  x )
36 n0 3794 . . . . . . . . . 10  |-  ( X  =/=  (/)  <->  E. p  p  e.  X )
3736biimpi 194 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. p  p  e.  X )
3837adantr 465 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  E. p  p  e.  X )
39 opelresi 5283 . . . . . . . . . . 11  |-  ( p  e.  X  ->  ( <. p ,  p >.  e.  (  _I  |`  X )  <-> 
p  e.  X ) )
4039ibir 242 . . . . . . . . . 10  |-  ( p  e.  X  ->  <. p ,  p >.  e.  (  _I  |`  X ) )
41 ne0i 3791 . . . . . . . . . 10  |-  ( <.
p ,  p >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
4240, 41syl 16 . . . . . . . . 9  |-  ( p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4342exlimiv 1698 . . . . . . . 8  |-  ( E. p  p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4438, 43syl 16 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (  _I  |`  X )  =/=  (/) )
4544adantr 465 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  (  _I  |`  X )  =/=  (/) )
46 ssn0 3818 . . . . . 6  |-  ( ( (  _I  |`  X ) 
C_  x  /\  (  _I  |`  X )  =/=  (/) )  ->  x  =/=  (/) )
4735, 45, 46syl2anc 661 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  x  =/=  (/) )
4847nelrdva 3313 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  -.  (/)  e.  F
)
49 df-nel 2665 . . . 4  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
5048, 49sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (/)  e/  F
)
51 df-ss 3490 . . . . . . . . 9  |-  ( x 
C_  y  <->  ( x  i^i  y )  =  x )
5251biimpi 194 . . . . . . . 8  |-  ( x 
C_  y  ->  (
x  i^i  y )  =  x )
5352adantl 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 )
5553, 54eqeltrd 2555 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  x  C_  y )  ->  (
x  i^i  y )  e.  F )
56 dfss1 3703 . . . . . . . . 9  |-  ( y 
C_  x  <->  ( x  i^i  y )  =  y )
5756biimpi 194 . . . . . . . 8  |-  ( y 
C_  x  ->  (
x  i^i  y )  =  y )
5857adantl 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 )
6058, 59eqeltrd 2555 . . . . . 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 )
641metustto 20796 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  F  /\  y  e.  F )  ->  (
x  C_  y  \/  y  C_  x ) )
6561, 62, 63, 64syl3anc 1228 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
( x  C_  y  \/  y  C_  x ) )
6655, 60, 65mpjaodan 784 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
( x  i^i  y
)  e.  F )
67 vex 3116 . . . . . . . . 9  |-  x  e. 
_V
6867inex1 4588 . . . . . . . 8  |-  ( x  i^i  y )  e. 
_V
6968pwid 4024 . . . . . . 7  |-  ( x  i^i  y )  e. 
~P ( x  i^i  y )
7069a1i 11 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
( x  i^i  y
)  e.  ~P (
x  i^i  y )
)
7170elpwid 4020 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
( x  i^i  y
)  C_  ( x  i^i  y ) )
72 sseq1 3525 . . . . . 6  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
7372rspcev 3214 . . . . 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 ) )
7466, 71, 73syl2anc 661 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  ->  E. z  e.  F  z  C_  ( x  i^i  y ) )
7574ralrimivva 2885 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y
) )
7633, 50, 753jca 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 5891 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
7877adantl 466 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  X  e.  _V )
79 xpexg 6709 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
8078, 78, 79syl2anc 661 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( X  X.  X )  e.  _V )
81 isfbas2 20071 . . 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 ) ) ) ) )
8280, 81syl 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 ) ) ) ) )
8317, 76, 82mpbir2and 920 1  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  F  e.  ( fBas `  ( X  X.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662    e/ wnel 2663   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   <.cop 4033    |-> cmpt 4505    _I cid 4790    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   -->wf 5582   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489   RR*cxr 9623   RR+crp 11216   [,)cico 11527  PsMetcpsmet 18173   fBascfbas 18177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-rp 11217  df-ico 11531  df-psmet 18182  df-fbas 18187
This theorem is referenced by:  metust  20806  cfilucfil  20808  metuel  20816  psmetutop  20821  restmetu  20825  metucn  20827
  Copyright terms: Public domain W3C validator