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Theorem metustfbasOLD 20143
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 20129 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  F  <->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) ) )
3 simpr 461 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  x  =  ( `' D " ( 0 [,) a ) ) )
4 cnvimass 5192 . . . . . . . . . 10  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
5 xmetf 19907 . . . . . . . . . . . 12  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
6 fdm 5566 . . . . . . . . . . . 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 465 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  dom  D  =  ( X  X.  X
) )
94, 8syl5sseq 3407 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  x  =  ( `' D " ( 0 [,) a ) ) )  ->  ( `' D " ( 0 [,) a ) )  C_  ( X  X.  X
) )
103, 9eqsstrd 3393 . . . . . . . 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 2847 . . . . . 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 2801 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  A. x  e.  F  x  C_  ( X  X.  X ) )
15 pwssb 4260 . . . 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 466 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  F  C_  ~P ( X  X.  X ) )
18 cnvexg 6527 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
19 imaexg 6518 . . . . . . . 8  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) 1 ) )  e.  _V )
20 elisset 2986 . . . . . . . 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 10998 . . . . . . . . 9  |-  1  e.  RR+
23 oveq2 6102 . . . . . . . . . . . 12  |-  ( a  =  1  ->  (
0 [,) a )  =  ( 0 [,) 1 ) )
2423imaeq2d 5172 . . . . . . . . . . 11  |-  ( a  =  1  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) 1
) ) )
2524eqeq2d 2454 . . . . . . . . . 10  |-  ( a  =  1  ->  (
x  =  ( `' D " ( 0 [,) a ) )  <-> 
x  =  ( `' D " ( 0 [,) 1 ) ) ) )
2625rspcev 3076 . . . . . . . . 9  |-  ( ( 1  e.  RR+  /\  x  =  ( `' D " ( 0 [,) 1
) ) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a
) ) )
2722, 26mpan 670 . . . . . . . 8  |-  ( x  =  ( `' D " ( 0 [,) 1
) )  ->  E. a  e.  RR+  x  =  ( `' D " ( 0 [,) a ) ) )
2827eximi 1625 . . . . . . 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 1680 . . . . . 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 466 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  E. x  x  e.  F )
33 n0 3649 . . . 4  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
3432, 33sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  F  =/=  (/) )
351metustidOLD 20137 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  F
)  ->  (  _I  |`  X )  C_  x
)
3635adantll 713 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  x  e.  F )  ->  (  _I  |`  X ) 
C_  x )
37 n0 3649 . . . . . . . . . 10  |-  ( X  =/=  (/)  <->  E. p  p  e.  X )
3837biimpi 194 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. p  p  e.  X )
3938adantr 465 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  E. p  p  e.  X )
40 opelresi 5125 . . . . . . . . . . 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 3646 . . . . . . . . . 10  |-  ( <.
p ,  p >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
4341, 42syl 16 . . . . . . . . 9  |-  ( p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4443exlimiv 1688 . . . . . . . 8  |-  ( E. p  p  e.  X  ->  (  _I  |`  X )  =/=  (/) )
4539, 44syl 16 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(  _I  |`  X )  =/=  (/) )
4645adantr 465 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  x  e.  F )  ->  (  _I  |`  X )  =/=  (/) )
47 ssn0 3673 . . . . . 6  |-  ( ( (  _I  |`  X ) 
C_  x  /\  (  _I  |`  X )  =/=  (/) )  ->  x  =/=  (/) )
4836, 46, 47syl2anc 661 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  x  e.  F )  ->  x  =/=  (/) )
4948nelrdva 3171 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  -.  (/)  e.  F )
50 df-nel 2612 . . . 4  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
5149, 50sylibr 212 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  (/) 
e/  F )
52 df-ss 3345 . . . . . . . . 9  |-  ( x 
C_  y  <->  ( x  i^i  y )  =  x )
5352biimpi 194 . . . . . . . 8  |-  ( x 
C_  y  ->  (
x  i^i  y )  =  x )
5453adantl 466 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  ( x  i^i  y )  =  x )
55 simplrl 759 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  x  e.  F )
5654, 55eqeltrd 2517 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  x  C_  y
)  ->  ( x  i^i  y )  e.  F
)
57 dfss1 3558 . . . . . . . . 9  |-  ( y 
C_  x  <->  ( x  i^i  y )  =  y )
5857biimpi 194 . . . . . . . 8  |-  ( y 
C_  x  ->  (
x  i^i  y )  =  y )
5958adantl 466 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  (
x  i^i  y )  =  y )
60 simplrr 760 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  y  e.  F )
6159, 60eqeltrd 2517 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  (
x  e.  F  /\  y  e.  F )
)  /\  y  C_  x )  ->  (
x  i^i  y )  e.  F )
62 simplr 754 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  D  e.  ( *Met `  X ) )
63 simprl 755 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  x  e.  F )
64 simprr 756 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  y  e.  F )
651metusttoOLD 20135 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  F  /\  y  e.  F
)  ->  ( x  C_  y  \/  y  C_  x ) )
6662, 63, 64, 65syl3anc 1218 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  (
x  C_  y  \/  y  C_  x ) )
6756, 61, 66mpjaodan 784 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  (
x  i^i  y )  e.  F )
68 vex 2978 . . . . . . . . 9  |-  x  e. 
_V
6968inex1 4436 . . . . . . . 8  |-  ( x  i^i  y )  e. 
_V
7069pwid 3877 . . . . . . 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 3873 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  (
x  i^i  y )  C_  ( x  i^i  y
) )
73 sseq1 3380 . . . . . 6  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
7473rspcev 3076 . . . . 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 661 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( x  e.  F  /\  y  e.  F
) )  ->  E. z  e.  F  z  C_  ( x  i^i  y
) )
7675ralrimivva 2811 . . 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 1168 . 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 5720 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  X  e.  _V )
7978adantl 466 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  X  e.  _V )
80 xpexg 6510 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
8179, 79, 80syl2anc 661 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( X  X.  X
)  e.  _V )
82 isfbas2 19411 . . 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 913 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 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2609    e/ wnel 2610   A.wral 2718   E.wrex 2719   _Vcvv 2975    i^i cin 3330    C_ wss 3331   (/)c0 3640   ~Pcpw 3863   <.cop 3886    e. cmpt 4353    _I cid 4634    X. cxp 4841   `'ccnv 4842   dom cdm 4843   ran crn 4844    |` cres 4845   "cima 4846   -->wf 5417   ` cfv 5421  (class class class)co 6094   0cc0 9285   1c1 9286   RR*cxr 9420   RR+crp 10994   [,)cico 11305   *Metcxmt 17804   fBascfbas 17807
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 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
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 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-po 4644  df-so 4645  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-1st 6580  df-2nd 6581  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-rp 10995  df-ico 11309  df-xmet 17813  df-fbas 17817
This theorem is referenced by:  metustOLD  20145  cfilucfilOLD  20147  metuelOLD  20155  metutopOLD  20160  metucnOLD  20166
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