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Theorem metustfbasOLD 18548
Description: The filter base generated by a metric  D. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustfbasOLD  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  F  e.  ( fBas `  ( X  X.  X ) ) )
Distinct variable groups:    D, a    X, a    F, a

Proof of Theorem metustfbasOLD
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
) ) )
21metustelOLD 18534 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  (
x  e.  F  <->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3 simpr 448 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  x  =  ( `' D " ( 0 [,) a ) ) )
4 cnvimass 5183 . . . . . . . . . 10  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
5 xmetf 18312 . . . . . . . . . . . 12  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
6 fdm 5554 . . . . . . . . . . . 12  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
75, 6syl 16 . . . . . . . . . . 11  |-  ( D  e.  ( * Met `  X )  ->  dom  D  =  ( X  X.  X ) )
87adantr 452 . . . . . . . . . 10  |-  ( ( D  e.  ( * Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  dom  D  =  ( X  X.  X
) )
94, 8syl5sseq 3356 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  ( `' D " ( 0 [,) a ) )  C_  ( X  X.  X
) )
103, 9eqsstrd 3342 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  x  C_  ( X  X.  X ) )
1110ex 424 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  (
x  =  ( `' D " ( 0 [,) a ) )  ->  x  C_  ( X  X.  X ) ) )
1211rexlimdvw 2793 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  ( E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) )  ->  x  C_  ( X  X.  X
) ) )
132, 12sylbid 207 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  (
x  e.  F  ->  x  C_  ( X  X.  X ) ) )
1413ralrimiv 2748 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  A. x  e.  F  x  C_  ( X  X.  X ) )
15 pwssb 4137 . . . 4  |-  ( F 
C_  ~P ( X  X.  X )  <->  A. x  e.  F  x  C_  ( X  X.  X ) )
1614, 15sylibr 204 . . 3  |-  ( D  e.  ( * Met `  X )  ->  F  C_ 
~P ( X  X.  X ) )
1716adantl 453 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  F  C_ 
~P ( X  X.  X ) )
18 cnvexg 5364 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  `' D  e.  _V )
19 imaexg 5176 . . . . . . . 8  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) 1 ) )  e.  _V )
20 elisset 2926 . . . . . . . 8  |-  ( ( `' D " ( 0 [,) 1 ) )  e.  _V  ->  E. x  x  =  ( `' D " ( 0 [,) 1 ) ) )
2118, 19, 203syl 19 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  E. x  x  =  ( `' D " ( 0 [,) 1 ) ) )
22 1rp 10572 . . . . . . . . 9  |-  1  e.  RR+
23 oveq2 6048 . . . . . . . . . . . 12  |-  ( a  =  1  ->  (
0 [,) a )  =  ( 0 [,) 1 ) )
2423imaeq2d 5162 . . . . . . . . . . 11  |-  ( a  =  1  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) 1
) ) )
2524eqeq2d 2415 . . . . . . . . . 10  |-  ( a  =  1  ->  (
x  =  ( `' D " ( 0 [,) a ) )  <-> 
x  =  ( `' D " ( 0 [,) 1 ) ) ) )
2625rspcev 3012 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  x  =  ( `' D " ( 0 [,) 1
) ) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
2722, 26mpan 652 . . . . . . . 8  |-  ( x  =  ( `' D " ( 0 [,) 1
) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2827eximi 1582 . . . . . . 7  |-  ( E. x  x  =  ( `' D " ( 0 [,) 1 ) )  ->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2921, 28syl 16 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
302exbidv 1633 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  ( E. x  x  e.  F 
<->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3129, 30mpbird 224 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  E. x  x  e.  F )
3231adantl 453 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  E. x  x  e.  F )
33 n0 3597 . . . 4  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
3432, 33sylibr 204 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  F  =/=  (/) )
351metustidOLD 18542 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  F
)  ->  (  _I  |`  X )  C_  x
)
3635adantll 695 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  F )  ->  (  _I  |`  X )  C_  x )
37 n0 3597 . . . . . . . . . 10  |-  ( X  =/=  (/)  <->  E. p  p  e.  X )
3837biimpi 187 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. p  p  e.  X )
3938adantr 452 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  E. p  p  e.  X )
40 opelresi 5117 . . . . . . . . . . 11  |-  ( p  e.  X  ->  ( <. p ,  p >.  e.  (  _I  |`  X )  <-> 
p  e.  X ) )
4140ibir 234 . . . . . . . . . 10  |-  ( p  e.  X  ->  <. p ,  p >.  e.  (  _I  |`  X ) )
42 ne0i 3594 . . . . . . . . . 10  |-  ( <.
p ,  p >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
4341, 42syl 16 . . . . . . . . 9  |-  ( p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4443exlimiv 1641 . . . . . . . 8  |-  ( E. p  p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4539, 44syl 16 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  (  _I  |`  X )  =/=  (/) )
4645adantr 452 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  F )  ->  (  _I  |`  X )  =/=  (/) )
47 ssn0 3620 . . . . . 6  |-  ( ( (  _I  |`  X ) 
C_  x  /\  (  _I  |`  X )  =/=  (/) )  ->  x  =/=  (/) )
4836, 46, 47syl2anc 643 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  F )  ->  x  =/=  (/) )
4948nelrdva 3103 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  -.  (/) 
e.  F )
50 df-nel 2570 . . . 4  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
5149, 50sylibr 204 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  (/)  e/  F
)
52 df-ss 3294 . . . . . . . . 9  |-  ( x 
C_  y  <->  ( x  i^i  y )  =  x )
5352biimpi 187 . . . . . . . 8  |-  ( x 
C_  y  ->  (
x  i^i  y )  =  x )
5453adantl 453 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  ( x  i^i  y )  =  x )
55 simplrl 737 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  x  e.  F )
5654, 55eqeltrd 2478 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  ( x  i^i  y )  e.  F
)
57 dfss1 3505 . . . . . . . . 9  |-  ( y 
C_  x  <->  ( x  i^i  y )  =  y )
5857biimpi 187 . . . . . . . 8  |-  ( y 
C_  x  ->  (
x  i^i  y )  =  y )
5958adantl 453 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  (
x  i^i  y )  =  y )
60 simplrr 738 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  y  e.  F )
6159, 60eqeltrd 2478 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  (
x  i^i  y )  e.  F )
62 simplr 732 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  ->  D  e.  ( * Met `  X
) )
63 simprl 733 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  ->  x  e.  F )
64 simprr 734 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  ->  y  e.  F )
651metusttoOLD 18540 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  F  /\  y  e.  F
)  ->  ( x  C_  y  \/  y  C_  x ) )
6662, 63, 64, 65syl3anc 1184 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  ->  ( x  C_  y  \/  y  C_  x ) )
6756, 61, 66mpjaodan 762 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  ->  ( x  i^i  y )  e.  F
)
68 vex 2919 . . . . . . . . 9  |-  x  e. 
_V
6968inex1 4304 . . . . . . . 8  |-  ( x  i^i  y )  e. 
_V
7069pwid 3772 . . . . . . 7  |-  ( x  i^i  y )  e. 
~P ( x  i^i  y )
7170a1i 11 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  ->  ( x  i^i  y )  e.  ~P ( x  i^i  y
) )
7271elpwid 3768 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  ->  ( x  i^i  y )  C_  (
x  i^i  y )
)
73 sseq1 3329 . . . . . 6  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
7473rspcev 3012 . . . . 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 ) )
7567, 72, 74syl2anc 643 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  ->  E. z  e.  F  z  C_  ( x  i^i  y
) )
7675ralrimivva 2758 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y
) )
7734, 51, 763jca 1134 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y ) ) )
78 elfvex 5717 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  X  e.  _V )
7978adantl 453 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  X  e.  _V )
80 xpexg 4948 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
8179, 79, 80syl2anc 643 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ( X  X.  X )  e. 
_V )
82 isfbas2 17820 . . 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 ) ) ) ) )
8381, 82syl 16 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) ) ) )
8417, 77, 83mpbir2and 889 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  F  e.  ( fBas `  ( X  X.  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567    e/ wnel 2568   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   <.cop 3777    e. cmpt 4226    _I cid 4453    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947   RR*cxr 9075   RR+crp 10568   [,)cico 10874   * Metcxmt 16641   fBascfbas 16644
This theorem is referenced by:  metustOLD  18550  cfilucfilOLD  18552  metuelOLD  18560  metutopOLD  18565  metucnOLD  18571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-rp 10569  df-ico 10878  df-xmet 16650  df-fbas 16654
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