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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.
StepHypRef Expression
1 metust.1 . . . . . . 7  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
21metustel 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 ) )
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 3547 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) a
) )  C_  ( X  X.  X ) )
103, 9eqsstrd 3533 . . . . . . . 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 2952 . . . . . 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 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 ) )
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 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 ) )
2322imaeq2d 5347 . . . . . . . . . . 11  |-  ( a  =  1  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) 1
) ) )
2423eqeq2d 2471 . . . . . . . . . 10  |-  ( a  =  1  ->  (
x  =  ( `' D " ( 0 [,) a ) )  <-> 
x  =  ( `' D " ( 0 [,) 1 ) ) ) )
2524rspcev 3210 . . . . . . . . 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 1657 . . . . . . 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 1715 . . . . . 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 3803 . . . 4  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
3331, 32sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  F  =/=  (/) )
341metustid 21188 . . . . . . 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 3803 . . . . . . . . . 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 5295 . . . . . . . . . . 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 3799 . . . . . . . . . 10  |-  ( <.
p ,  p >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
4240, 41syl 16 . . . . . . . . 9  |-  ( p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4342exlimiv 1723 . . . . . . . 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 3827 . . . . . 6  |-  ( ( (  _I  |`  X ) 
C_  x  /\  (  _I  |`  X )  =/=  (/) )  ->  x  =/=  (/) )
4735, 45, 46syl2anc 661 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  x  =/=  (/) )
4847nelrdva 3309 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  -.  (/)  e.  F
)
49 df-nel 2655 . . . 4  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
5048, 49sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (/)  e/  F
)
51 df-ss 3485 . . . . . . . . 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 761 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  x  C_  y )  ->  x  e.  F )
5553, 54eqeltrd 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 )
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 762 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  y  C_  x )  ->  y  e.  F )
6058, 59eqeltrd 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 )
641metustto 21186 . . . . . . 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 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
6867inex1 4597 . . . . . . . 8  |-  ( x  i^i  y )  e. 
_V
6968pwid 4029 . . . . . . 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 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 )
) )
7372rspcev 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 ) )
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 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
) )
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 5899 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
7877adantl 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 )
8078, 78, 79syl2anc 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 ) ) ) ) )
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 922 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 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
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