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Theorem metustfbas 21621
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 21614 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  F  <->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3 simpr 467 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  ->  x  =  ( `' D " ( 0 [,) a ) ) )
4 cnvimass 5207 . . . . . . . . . 10  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
5 psmetf 21371 . . . . . . . . . . . 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 471 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  ->  dom  D  =  ( X  X.  X ) )
94, 8syl5sseq 3492 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) a
) )  C_  ( X  X.  X ) )
103, 9eqsstrd 3478 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  x  =  ( `' D " ( 0 [,) a
) ) )  ->  x  C_  ( X  X.  X ) )
1110ex 440 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  ( x  =  ( `' D " ( 0 [,) a
) )  ->  x  C_  ( X  X.  X
) ) )
1211rexlimdvw 2894 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( E. a  e.  RR+  x  =  ( `' D "
( 0 [,) a
) )  ->  x  C_  ( X  X.  X
) ) )
132, 12sylbid 223 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  F  ->  x  C_  ( X  X.  X
) ) )
1413ralrimiv 2812 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  F  x  C_  ( X  X.  X ) )
15 pwssb 4382 . . . 4  |-  ( F 
C_  ~P ( X  X.  X )  <->  A. x  e.  F  x  C_  ( X  X.  X ) )
1614, 15sylibr 217 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  F  C_  ~P ( X  X.  X
) )
1716adantl 472 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  F  C_  ~P ( X  X.  X
) )
18 cnvexg 6766 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
19 imaexg 6757 . . . . . . 7  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) 1 ) )  e.  _V )
20 elisset 3069 . . . . . . 7  |-  ( ( `' D " ( 0 [,) 1 ) )  e.  _V  ->  E. x  x  =  ( `' D " ( 0 [,) 1 ) ) )
21 1rp 11335 . . . . . . . . 9  |-  1  e.  RR+
22 oveq2 6323 . . . . . . . . . . . 12  |-  ( a  =  1  ->  (
0 [,) a )  =  ( 0 [,) 1 ) )
2322imaeq2d 5187 . . . . . . . . . . 11  |-  ( a  =  1  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) 1
) ) )
2423eqeq2d 2472 . . . . . . . . . 10  |-  ( a  =  1  ->  (
x  =  ( `' D " ( 0 [,) a ) )  <-> 
x  =  ( `' D " ( 0 [,) 1 ) ) ) )
2524rspcev 3162 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  x  =  ( `' D " ( 0 [,) 1
) ) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
2621, 25mpan 681 . . . . . . . 8  |-  ( x  =  ( `' D " ( 0 [,) 1
) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2726eximi 1718 . . . . . . 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 1779 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( E. x  x  e.  F  <->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3028, 29mpbird 240 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  E. x  x  e.  F )
3130adantl 472 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  E. x  x  e.  F )
32 n0 3753 . . . 4  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
3331, 32sylibr 217 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  F  =/=  (/) )
341metustid 21618 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  F )  ->  (  _I  |`  X )  C_  x )
3534adantll 725 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  (  _I  |`  X )  C_  x )
36 n0 3753 . . . . . . . . . 10  |-  ( X  =/=  (/)  <->  E. p  p  e.  X )
3736biimpi 199 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. p  p  e.  X )
3837adantr 471 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  E. p  p  e.  X )
39 opelresi 5135 . . . . . . . . . . 11  |-  ( p  e.  X  ->  ( <. p ,  p >.  e.  (  _I  |`  X )  <-> 
p  e.  X ) )
4039ibir 250 . . . . . . . . . 10  |-  ( p  e.  X  ->  <. p ,  p >.  e.  (  _I  |`  X ) )
41 ne0i 3749 . . . . . . . . . 10  |-  ( <.
p ,  p >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
4240, 41syl 17 . . . . . . . . 9  |-  ( p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4342exlimiv 1787 . . . . . . . 8  |-  ( E. p  p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4438, 43syl 17 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (  _I  |`  X )  =/=  (/) )
4544adantr 471 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  (  _I  |`  X )  =/=  (/) )
46 ssn0 3779 . . . . . 6  |-  ( ( (  _I  |`  X ) 
C_  x  /\  (  _I  |`  X )  =/=  (/) )  ->  x  =/=  (/) )
4735, 45, 46syl2anc 671 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  x  e.  F )  ->  x  =/=  (/) )
4847nelrdva 3261 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  -.  (/)  e.  F
)
49 df-nel 2636 . . . 4  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
5048, 49sylibr 217 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (/)  e/  F
)
51 df-ss 3430 . . . . . . . . 9  |-  ( x 
C_  y  <->  ( x  i^i  y )  =  x )
5251biimpi 199 . . . . . . . 8  |-  ( x 
C_  y  ->  (
x  i^i  y )  =  x )
5352adantl 472 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  x  C_  y )  ->  (
x  i^i  y )  =  x )
54 simplrl 775 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  x  C_  y )  ->  x  e.  F )
5553, 54eqeltrd 2540 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  x  C_  y )  ->  (
x  i^i  y )  e.  F )
56 dfss1 3649 . . . . . . . . 9  |-  ( y 
C_  x  <->  ( x  i^i  y )  =  y )
5756biimpi 199 . . . . . . . 8  |-  ( y 
C_  x  ->  (
x  i^i  y )  =  y )
5857adantl 472 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  y  C_  x )  ->  (
x  i^i  y )  =  y )
59 simplrr 776 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  y  C_  x )  ->  y  e.  F )
6058, 59eqeltrd 2540 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  /\  y  C_  x )  ->  (
x  i^i  y )  e.  F )
61 simplr 767 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  ->  D  e.  (PsMet `  X
) )
62 simprl 769 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  ->  x  e.  F )
63 simprr 771 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
y  e.  F )
641metustto 21617 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  F  /\  y  e.  F )  ->  (
x  C_  y  \/  y  C_  x ) )
6561, 62, 63, 64syl3anc 1276 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
( x  C_  y  \/  y  C_  x ) )
6655, 60, 65mpjaodan 800 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
( x  i^i  y
)  e.  F )
67 vex 3060 . . . . . . . . 9  |-  x  e. 
_V
6867inex1 4558 . . . . . . . 8  |-  ( x  i^i  y )  e. 
_V
6968pwid 3977 . . . . . . 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 3973 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  -> 
( x  i^i  y
)  C_  ( x  i^i  y ) )
72 sseq1 3465 . . . . . 6  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
7372rspcev 3162 . . . . 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 671 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( x  e.  F  /\  y  e.  F ) )  ->  E. z  e.  F  z  C_  ( x  i^i  y ) )
7574ralrimivva 2821 . . 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 1194 . 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 5915 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
7877adantl 472 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  X  e.  _V )
79 xpexg 6620 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
8078, 78, 79syl2anc 671 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( X  X.  X )  e.  _V )
81 isfbas2 20899 . . 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 938 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 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898    =/= wne 2633    e/ wnel 2634   A.wral 2749   E.wrex 2750   _Vcvv 3057    i^i cin 3415    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   <.cop 3986    |-> cmpt 4475    _I cid 4763    X. cxp 4851   `'ccnv 4852   dom cdm 4853   ran crn 4854    |` cres 4855   "cima 4856   -->wf 5597   ` cfv 5601  (class class class)co 6315   0cc0 9565   1c1 9566   RR*cxr 9700   RR+crp 11331   [,)cico 11666  PsMetcpsmet 19003   fBascfbas 19007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-rp 11332  df-ico 11670  df-psmet 19011  df-fbas 19016
This theorem is referenced by:  metust  21622  cfilucfil  21623  metuel  21628  psmetutop  21631  restmetu  21634  metucn  21635
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