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Theorem metustfbasOLD 19984
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 19970 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  F  <->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3 simpr 458 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  x  =  ( `' D " ( 0 [,) a ) ) )
4 cnvimass 5179 . . . . . . . . . 10  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
5 xmetf 19748 . . . . . . . . . . . 12  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
6 fdm 5553 . . . . . . . . . . . 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 462 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  dom  D  =  ( X  X.  X
) )
94, 8syl5sseq 3394 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  ( `' D " ( 0 [,) a ) )  C_  ( X  X.  X
) )
103, 9eqsstrd 3380 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  x  C_  ( X  X.  X ) )
1110ex 434 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  (
x  =  ( `' D " ( 0 [,) a ) )  ->  x  C_  ( X  X.  X ) ) )
1211rexlimdvw 2836 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  ( E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) )  ->  x  C_  ( X  X.  X
) ) )
132, 12sylbid 215 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  F  ->  x  C_  ( X  X.  X ) ) )
1413ralrimiv 2790 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  A. x  e.  F  x  C_  ( X  X.  X ) )
15 pwssb 4247 . . . 4  |-  ( F 
C_  ~P ( X  X.  X )  <->  A. x  e.  F  x  C_  ( X  X.  X ) )
1614, 15sylibr 212 . . 3  |-  ( D  e.  ( *Met `  X )  ->  F  C_ 
~P ( X  X.  X ) )
1716adantl 463 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  F  C_  ~P ( X  X.  X ) )
18 cnvexg 6515 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
19 imaexg 6506 . . . . . . . 8  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) 1 ) )  e.  _V )
20 elisset 2975 . . . . . . . 8  |-  ( ( `' D " ( 0 [,) 1 ) )  e.  _V  ->  E. x  x  =  ( `' D " ( 0 [,) 1 ) ) )
2118, 19, 203syl 20 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  E. x  x  =  ( `' D " ( 0 [,) 1 ) ) )
22 1rp 10985 . . . . . . . . 9  |-  1  e.  RR+
23 oveq2 6090 . . . . . . . . . . . 12  |-  ( a  =  1  ->  (
0 [,) a )  =  ( 0 [,) 1 ) )
2423imaeq2d 5159 . . . . . . . . . . 11  |-  ( a  =  1  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) 1
) ) )
2524eqeq2d 2446 . . . . . . . . . 10  |-  ( a  =  1  ->  (
x  =  ( `' D " ( 0 [,) a ) )  <-> 
x  =  ( `' D " ( 0 [,) 1 ) ) ) )
2625rspcev 3064 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  x  =  ( `' D " ( 0 [,) 1
) ) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
2722, 26mpan 665 . . . . . . . 8  |-  ( x  =  ( `' D " ( 0 [,) 1
) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2827eximi 1628 . . . . . . 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 1681 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  ( E. x  x  e.  F 
<->  E. x E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3129, 30mpbird 232 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  E. x  x  e.  F )
3231adantl 463 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  E. x  x  e.  F )
33 n0 3636 . . . 4  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
3432, 33sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  F  =/=  (/) )
351metustidOLD 19978 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  F
)  ->  (  _I  |`  X )  C_  x
)
3635adantll 708 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  x  e.  F )  ->  (  _I  |`  X ) 
C_  x )
37 n0 3636 . . . . . . . . . 10  |-  ( X  =/=  (/)  <->  E. p  p  e.  X )
3837biimpi 194 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. p  p  e.  X )
3938adantr 462 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  E. p  p  e.  X )
40 opelresi 5112 . . . . . . . . . . 11  |-  ( p  e.  X  ->  ( <. p ,  p >.  e.  (  _I  |`  X )  <-> 
p  e.  X ) )
4140ibir 242 . . . . . . . . . 10  |-  ( p  e.  X  ->  <. p ,  p >.  e.  (  _I  |`  X ) )
42 ne0i 3633 . . . . . . . . . 10  |-  ( <.
p ,  p >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
4341, 42syl 16 . . . . . . . . 9  |-  ( p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4443exlimiv 1689 . . . . . . . 8  |-  ( E. p  p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4539, 44syl 16 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(  _I  |`  X )  =/=  (/) )
4645adantr 462 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  x  e.  F )  ->  (  _I  |`  X )  =/=  (/) )
47 ssn0 3660 . . . . . 6  |-  ( ( (  _I  |`  X ) 
C_  x  /\  (  _I  |`  X )  =/=  (/) )  ->  x  =/=  (/) )
4836, 46, 47syl2anc 656 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  x  e.  F )  ->  x  =/=  (/) )
4948nelrdva 3159 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  -.  (/)  e.  F )
50 df-nel 2601 . . . 4  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
5149, 50sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  (/) 
e/  F )
52 df-ss 3332 . . . . . . . . 9  |-  ( x 
C_  y  <->  ( x  i^i  y )  =  x )
5352biimpi 194 . . . . . . . 8  |-  ( x 
C_  y  ->  (
x  i^i  y )  =  x )
5453adantl 463 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  ( x  i^i  y )  =  x )
55 simplrl 754 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  x  e.  F )
5654, 55eqeltrd 2509 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  ( x  i^i  y )  e.  F
)
57 dfss1 3545 . . . . . . . . 9  |-  ( y 
C_  x  <->  ( x  i^i  y )  =  y )
5857biimpi 194 . . . . . . . 8  |-  ( y 
C_  x  ->  (
x  i^i  y )  =  y )
5958adantl 463 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  (
x  i^i  y )  =  y )
60 simplrr 755 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  y  e.  F )
6159, 60eqeltrd 2509 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  (
x  i^i  y )  e.  F )
62 simplr 749 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  D  e.  ( *Met `  X ) )
63 simprl 750 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  x  e.  F )
64 simprr 751 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  y  e.  F )
651metusttoOLD 19976 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  F  /\  y  e.  F
)  ->  ( x  C_  y  \/  y  C_  x ) )
6662, 63, 64, 65syl3anc 1213 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  (
x  C_  y  \/  y  C_  x ) )
6756, 61, 66mpjaodan 779 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  (
x  i^i  y )  e.  F )
68 vex 2967 . . . . . . . . 9  |-  x  e. 
_V
6968inex1 4423 . . . . . . . 8  |-  ( x  i^i  y )  e. 
_V
7069pwid 3864 . . . . . . 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 3860 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  (
x  i^i  y )  C_  ( x  i^i  y
) )
73 sseq1 3367 . . . . . 6  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
7473rspcev 3064 . . . . 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 656 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  E. z  e.  F  z  C_  ( x  i^i  y
) )
7675ralrimivva 2800 . . 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 1163 . 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 5707 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  X  e.  _V )
7978adantl 463 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  X  e.  _V )
80 xpexg 6498 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
8179, 79, 80syl2anc 656 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( X  X.  X
)  e.  _V )
82 isfbas2 19252 . . 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 908 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  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 960    = wceq 1364   E.wex 1591    e. wcel 1757    =/= wne 2598    e/ wnel 2599   A.wral 2707   E.wrex 2708   _Vcvv 2964    i^i cin 3317    C_ wss 3318   (/)c0 3627   ~Pcpw 3850   <.cop 3873    e. cmpt 4340    _I cid 4620    X. cxp 4827   `'ccnv 4828   dom cdm 4829   ran crn 4830    |` cres 4831   "cima 4832   -->wf 5404   ` cfv 5408  (class class class)co 6082   0cc0 9272   1c1 9273   RR*cxr 9407   RR+crp 10981   [,)cico 11292   *Metcxmt 17647   fBascfbas 17650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-po 4630  df-so 4631  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6568  df-2nd 6569  df-er 7091  df-map 7206  df-en 7301  df-dom 7302  df-sdom 7303  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-rp 10982  df-ico 11296  df-xmet 17656  df-fbas 17660
This theorem is referenced by:  metustOLD  19986  cfilucfilOLD  19988  metuelOLD  19996  metutopOLD  20001  metucnOLD  20007
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