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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.
StepHypRef Expression
1 metust.1 . . . . . . 7  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
21metustel 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 ) )
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 3409 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) a
) )  C_  ( X  X.  X ) )
103, 9eqsstrd 3395 . . . . . . . 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 2849 . . . . . 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 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 ) )
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 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 ) )
2322imaeq2d 5174 . . . . . . . . . . 11  |-  ( a  =  1  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) 1
) ) )
2423eqeq2d 2454 . . . . . . . . . 10  |-  ( a  =  1  ->  (
x  =  ( `' D " ( 0 [,) a ) )  <-> 
x  =  ( `' D " ( 0 [,) 1 ) ) ) )
2524rspcev 3078 . . . . . . . . 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 1625 . . . . . . 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 1680 . . . . . 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 3651 . . . 4  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
3331, 32sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  F  =/=  (/) )
341metustid 20140 . . . . . . 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 3651 . . . . . . . . . 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 5127 . . . . . . . . . . 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 3648 . . . . . . . . . 10  |-  ( <.
p ,  p >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
4240, 41syl 16 . . . . . . . . 9  |-  ( p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4342exlimiv 1688 . . . . . . . 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 3675 . . . . . 6  |-  ( ( (  _I  |`  X ) 
C_  x  /\  (  _I  |`  X )  =/=  (/) )  ->  x  =/=  (/) )
4735, 45, 46syl2anc 661 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  x  =/=  (/) )
4847nelrdva 3173 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  -.  (/)  e.  F
)
49 df-nel 2614 . . . 4  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
5048, 49sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (/)  e/  F
)
51 df-ss 3347 . . . . . . . . 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 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 )
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 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 )
641metustto 20138 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  F  /\  y  e.  F )  ->  (
x  C_  y  \/  y  C_  x ) )
6561, 62, 63, 64syl3anc 1218 . . . . . 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 2980 . . . . . . . . 9  |-  x  e. 
_V
6867inex1 4438 . . . . . . . 8  |-  ( x  i^i  y )  e. 
_V
6968pwid 3879 . . . . . . 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 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 )
) )
7372rspcev 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 ) )
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 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
) )
7633, 50, 753jca 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 )
7877adantl 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 )
8078, 78, 79syl2anc 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 ) ) ) ) )
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 913 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 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
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