MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metustsym Structured version   Unicode version

Theorem metustsym 20112
Description: Elements of the filter base generated by the metric  D are symmetric. (Contributed by Thierry Arnoux, 28-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
metustsym  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  `' A  =  A )
Distinct variable groups:    D, a    X, a    A, a    F, a

Proof of Theorem metustsym
Dummy variables  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metust.1 . . . 4  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
21metustss 20104 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X
) )
3 cnvss 5007 . . . 4  |-  ( A 
C_  ( X  X.  X )  ->  `' A  C_  `' ( X  X.  X ) )
4 cnvxp 5250 . . . 4  |-  `' ( X  X.  X )  =  ( X  X.  X )
53, 4syl6sseq 3397 . . 3  |-  ( A 
C_  ( X  X.  X )  ->  `' A  C_  ( X  X.  X ) )
62, 5syl 16 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  `' A  C_  ( X  X.  X ) )
7 simp-4l 765 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  D  e.  (PsMet `  X
) )
8 simpr1r 1046 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
( p  e.  X  /\  q  e.  X
)  /\  a  e.  RR+ 
/\  A  =  ( `' D " ( 0 [,) a ) ) ) )  ->  q  e.  X )
983anassrs 1209 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
q  e.  X )
10 simpr1l 1045 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
( p  e.  X  /\  q  e.  X
)  /\  a  e.  RR+ 
/\  A  =  ( `' D " ( 0 [,) a ) ) ) )  ->  p  e.  X )
11103anassrs 1209 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  p  e.  X )
12 psmetsym 19861 . . . . . . . . . . 11  |-  ( ( D  e.  (PsMet `  X )  /\  q  e.  X  /\  p  e.  X )  ->  (
q D p )  =  ( p D q ) )
137, 9, 11, 12syl3anc 1218 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( q D p )  =  ( p D q ) )
14 df-ov 6089 . . . . . . . . . 10  |-  ( q D p )  =  ( D `  <. q ,  p >. )
15 df-ov 6089 . . . . . . . . . 10  |-  ( p D q )  =  ( D `  <. p ,  q >. )
1613, 14, 153eqtr3g 2493 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( D `  <. q ,  p >. )  =  ( D `  <. p ,  q >.
) )
1716eleq1d 2504 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( ( D `  <. q ,  p >. )  e.  ( 0 [,) a )  <->  ( D `  <. p ,  q
>. )  e.  (
0 [,) a ) ) )
18 psmetf 19857 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
19 ffun 5556 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
207, 18, 193syl 20 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  Fun  D )
21 simpllr 758 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( p  e.  X  /\  q  e.  X
) )
2221ancomd 451 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( q  e.  X  /\  p  e.  X
) )
23 opelxpi 4866 . . . . . . . . . . 11  |-  ( ( q  e.  X  /\  p  e.  X )  -> 
<. q ,  p >.  e.  ( X  X.  X
) )
2422, 23syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. q ,  p >.  e.  ( X  X.  X
) )
25 fdm 5558 . . . . . . . . . . . 12  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
2618, 25syl 16 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
277, 26syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  dom  D  =  ( X  X.  X ) )
2824, 27eleqtrrd 2515 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. q ,  p >.  e. 
dom  D )
29 fvimacnv 5813 . . . . . . . . 9  |-  ( ( Fun  D  /\  <. q ,  p >.  e.  dom  D )  ->  ( ( D `  <. q ,  p >. )  e.  ( 0 [,) a )  <->  <. q ,  p >.  e.  ( `' D "
( 0 [,) a
) ) ) )
3020, 28, 29syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( ( D `  <. q ,  p >. )  e.  ( 0 [,) a )  <->  <. q ,  p >.  e.  ( `' D " ( 0 [,) a ) ) ) )
31 opelxpi 4866 . . . . . . . . . . 11  |-  ( ( p  e.  X  /\  q  e.  X )  -> 
<. p ,  q >.  e.  ( X  X.  X
) )
3221, 31syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. p ,  q >.  e.  ( X  X.  X
) )
3332, 27eleqtrrd 2515 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. p ,  q >.  e.  dom  D )
34 fvimacnv 5813 . . . . . . . . 9  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
3520, 33, 34syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( ( D `  <. p ,  q >.
)  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) ) )
3617, 30, 353bitr3d 283 . . . . . . 7  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( <. q ,  p >.  e.  ( `' D " ( 0 [,) a
) )  <->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) ) )
37 simpr 461 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  A  =  ( `' D " ( 0 [,) a ) ) )
3837eleq2d 2505 . . . . . . 7  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( <. q ,  p >.  e.  A  <->  <. q ,  p >.  e.  ( `' D " ( 0 [,) a ) ) ) )
3937eleq2d 2505 . . . . . . 7  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( <. p ,  q
>.  e.  A  <->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) ) )
4036, 38, 393bitr4d 285 . . . . . 6  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( <. q ,  p >.  e.  A  <->  <. p ,  q >.  e.  A
) )
41 simplr 754 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  A  e.  F )
42 eqid 2438 . . . . . . . . . 10  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
4342elrnmpt 5081 . . . . . . . . 9  |-  ( A  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  ->  ( A  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) ) )
4443ibi 241 . . . . . . . 8  |-  ( A  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4544, 1eleq2s 2530 . . . . . . 7  |-  ( A  e.  F  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4641, 45syl 16 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4740, 46r19.29a 2857 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  ( <. q ,  p >.  e.  A  <->  <.
p ,  q >.  e.  A ) )
48 df-br 4288 . . . . . 6  |-  ( p `' A q  <->  <. p ,  q >.  e.  `' A )
49 vex 2970 . . . . . . 7  |-  p  e. 
_V
50 vex 2970 . . . . . . 7  |-  q  e. 
_V
5149, 50opelcnv 5016 . . . . . 6  |-  ( <.
p ,  q >.  e.  `' A  <->  <. q ,  p >.  e.  A )
5248, 51bitri 249 . . . . 5  |-  ( p `' A q  <->  <. q ,  p >.  e.  A
)
53 df-br 4288 . . . . 5  |-  ( p A q  <->  <. p ,  q >.  e.  A
)
5447, 52, 533bitr4g 288 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  ( p `' A q  <->  p A
q ) )
5554ex 434 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  (
( p  e.  X  /\  q  e.  X
)  ->  ( p `' A q  <->  p A
q ) ) )
56553impib 1185 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  p  e.  X  /\  q  e.  X )  ->  (
p `' A q  <-> 
p A q ) )
576, 2, 56eqbrrdva 5004 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  `' A  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2711    C_ wss 3323   <.cop 3878   class class class wbr 4287    e. cmpt 4345    X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6086   0cc0 9274   RR*cxr 9409   RR+crp 10983   [,)cico 11294  PsMetcpsmet 17775
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-xadd 11082  df-psmet 17784
This theorem is referenced by:  metust  20118
  Copyright terms: Public domain W3C validator