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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.
StepHypRef Expression
1 metust.1 . . . . . . 7  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
21metustel 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 ) )
75, 6syl 17 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
87adantr 467 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  ->  dom  D  =  ( X  X.  X ) )
94, 8syl5sseq 3518 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) a
) )  C_  ( X  X.  X ) )
103, 9eqsstrd 3504 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  ->  x  C_  ( X  X.  X ) )
1110ex 436 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  ( x  =  ( `' D " ( 0 [,) a
) )  ->  x  C_  ( X  X.  X
) ) )
1211rexlimdvw 2922 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( E. a  e.  RR+  x  =  ( `' D "
( 0 [,) a
) )  ->  x  C_  ( X  X.  X
) ) )
132, 12sylbid 219 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  F  ->  x  C_  ( X  X.  X
) ) )
1413ralrimiv 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 ) )
1614, 15sylibr 216 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  F  C_  ~P ( X  X.  X
) )
1716adantl 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 ) )
2322imaeq2d 5193 . . . . . . . . . . 11  |-  ( a  =  1  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) 1
) ) )
2423eqeq2d 2437 . . . . . . . . . 10  |-  ( a  =  1  ->  (
x  =  ( `' D " ( 0 [,) a ) )  <-> 
x  =  ( `' D " ( 0 [,) 1 ) ) ) )
2524rspcev 3188 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  x  =  ( `' D " ( 0 [,) 1
) ) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
2621, 25mpan 675 . . . . . . . 8  |-  ( x  =  ( `' D " ( 0 [,) 1
) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2726eximi 1702 . . . . . . 7  |-  ( E. x  x  =  ( `' D " ( 0 [,) 1 ) )  ->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2818, 19, 20, 274syl 19 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
292exbidv 1763 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( E. x  x  e.  F  <->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3028, 29mpbird 236 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  E. x  x  e.  F )
3130adantl 468 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  E. x  x  e.  F )
32 n0 3777 . . . 4  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
3331, 32sylibr 216 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  F  =/=  (/) )
341metustid 21573 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  F )  ->  (  _I  |`  X )  C_  x )
3534adantll 719 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  (  _I  |`  X )  C_  x )
36 n0 3777 . . . . . . . . . 10  |-  ( X  =/=  (/)  <->  E. p  p  e.  X )
3736biimpi 198 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. p  p  e.  X )
3837adantr 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 ) )
4039ibir 246 . . . . . . . . . 10  |-  ( p  e.  X  ->  <. p ,  p >.  e.  (  _I  |`  X ) )
41 ne0i 3773 . . . . . . . . . 10  |-  ( <.
p ,  p >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
4240, 41syl 17 . . . . . . . . 9  |-  ( p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4342exlimiv 1771 . . . . . . . 8  |-  ( E. p  p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4438, 43syl 17 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (  _I  |`  X )  =/=  (/) )
4544adantr 467 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  (  _I  |`  X )  =/=  (/) )
46 ssn0 3803 . . . . . 6  |-  ( ( (  _I  |`  X ) 
C_  x  /\  (  _I  |`  X )  =/=  (/) )  ->  x  =/=  (/) )
4735, 45, 46syl2anc 666 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  x  =/=  (/) )
4847nelrdva 3287 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  -.  (/)  e.  F
)
49 df-nel 2622 . . . 4  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
5048, 49sylibr 216 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (/)  e/  F
)
51 df-ss 3456 . . . . . . . . 9  |-  ( x 
C_  y  <->  ( x  i^i  y )  =  x )
5251biimpi 198 . . . . . . . 8  |-  ( x 
C_  y  ->  (
x  i^i  y )  =  x )
5352adantl 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 )
5553, 54eqeltrd 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 )
5756biimpi 198 . . . . . . . 8  |-  ( y 
C_  x  ->  (
x  i^i  y )  =  y )
5857adantl 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 )
6058, 59eqeltrd 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 )
641metustto 21572 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  F  /\  y  e.  F )  ->  (
x  C_  y  \/  y  C_  x ) )
6561, 62, 63, 64syl3anc 1265 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
( x  C_  y  \/  y  C_  x ) )
6655, 60, 65mpjaodan 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
6867inex1 4571 . . . . . . . 8  |-  ( x  i^i  y )  e. 
_V
6968pwid 4001 . . . . . . 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 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 )
) )
7372rspcev 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 ) )
7466, 71, 73syl2anc 666 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  ->  E. z  e.  F  z  C_  ( x  i^i  y ) )
7574ralrimivva 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
) )
7633, 50, 753jca 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 )
7877adantl 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 )
8078, 78, 79syl2anc 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 ) ) ) ) )
8280, 81syl 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 ) ) ) ) )
8317, 76, 82mpbir2and 931 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 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
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