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Theorem metustsym 20268
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 20260 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X
) )
3 cnvss 5119 . . . 4  |-  ( A 
C_  ( X  X.  X )  ->  `' A  C_  `' ( X  X.  X ) )
4 cnvxp 5362 . . . 4  |-  `' ( X  X.  X )  =  ( X  X.  X )
53, 4syl6sseq 3509 . . 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 1210 . . . . . . . . . . 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 1210 . . . . . . . . . . 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 20017 . . . . . . . . . . 11  |-  ( ( D  e.  (PsMet `  X )  /\  q  e.  X  /\  p  e.  X )  ->  (
q D p )  =  ( p D q ) )
137, 9, 11, 12syl3anc 1219 . . . . . . . . . 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 6202 . . . . . . . . . 10  |-  ( q D p )  =  ( D `  <. q ,  p >. )
15 df-ov 6202 . . . . . . . . . 10  |-  ( p D q )  =  ( D `  <. p ,  q >. )
1613, 14, 153eqtr3g 2518 . . . . . . . . 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 2523 . . . . . . . 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 20013 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
19 ffun 5668 . . . . . . . . . 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 4978 . . . . . . . . . . 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 5670 . . . . . . . . . . . 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 2545 . . . . . . . . 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 5926 . . . . . . . . 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 4978 . . . . . . . . . . 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 2545 . . . . . . . . 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 5926 . . . . . . . . 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 2524 . . . . . . 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 2524 . . . . . . 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 2454 . . . . . . . . . 10  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
4342elrnmpt 5193 . . . . . . . . 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 2562 . . . . . . 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 2966 . . . . 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 4400 . . . . . 6  |-  ( p `' A q  <->  <. p ,  q >.  e.  `' A )
49 vex 3079 . . . . . . 7  |-  p  e. 
_V
50 vex 3079 . . . . . . 7  |-  q  e. 
_V
5149, 50opelcnv 5128 . . . . . 6  |-  ( <.
p ,  q >.  e.  `' A  <->  <. q ,  p >.  e.  A )
5248, 51bitri 249 . . . . 5  |-  ( p `' A q  <->  <. q ,  p >.  e.  A
)
53 df-br 4400 . . . . 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 1186 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  p  e.  X  /\  q  e.  X )  ->  (
p `' A q  <-> 
p A q ) )
576, 2, 56eqbrrdva 5116 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 1370    e. wcel 1758   E.wrex 2799    C_ wss 3435   <.cop 3990   class class class wbr 4399    |-> cmpt 4457    X. cxp 4945   `'ccnv 4946   dom cdm 4947   ran crn 4948   "cima 4950   Fun wfun 5519   -->wf 5521   ` cfv 5525  (class class class)co 6199   0cc0 9392   RR*cxr 9527   RR+crp 11101   [,)cico 11412  PsMetcpsmet 17924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-xadd 11200  df-psmet 17933
This theorem is referenced by:  metust  20274
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