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Theorem metustsym 21648
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 21644 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X
) )
3 cnvss 5012 . . . 4  |-  ( A 
C_  ( X  X.  X )  ->  `' A  C_  `' ( X  X.  X ) )
4 cnvxp 5260 . . . 4  |-  `' ( X  X.  X )  =  ( X  X.  X )
53, 4syl6sseq 3464 . . 3  |-  ( A 
C_  ( X  X.  X )  ->  `' A  C_  ( X  X.  X ) )
62, 5syl 17 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  `' A  C_  ( X  X.  X ) )
7 simp-4l 784 . . . . . . . . . 10  |-  ( ( ( ( ( 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 1088 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
( p  e.  X  /\  q  e.  X
)  /\  a  e.  RR+ 
/\  A  =  ( `' D " ( 0 [,) a ) ) ) )  ->  q  e.  X )
983anassrs 1256 . . . . . . . . . 10  |-  ( ( ( ( ( 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 1087 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
( p  e.  X  /\  q  e.  X
)  /\  a  e.  RR+ 
/\  A  =  ( `' D " ( 0 [,) a ) ) ) )  ->  p  e.  X )
11103anassrs 1256 . . . . . . . . . 10  |-  ( ( ( ( ( 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 21404 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  q  e.  X  /\  p  e.  X )  ->  (
q D p )  =  ( p D q ) )
137, 9, 11, 12syl3anc 1292 . . . . . . . . 9  |-  ( ( ( ( ( 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 6311 . . . . . . . . 9  |-  ( q D p )  =  ( D `  <. q ,  p >. )
15 df-ov 6311 . . . . . . . . 9  |-  ( p D q )  =  ( D `  <. p ,  q >. )
1613, 14, 153eqtr3g 2528 . . . . . . . 8  |-  ( ( ( ( ( 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 2533 . . . . . . 7  |-  ( ( ( ( ( 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 21400 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
19 ffun 5742 . . . . . . . . 9  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
207, 18, 193syl 18 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  Fun  D )
21 simpllr 777 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 458 . . . . . . . . . 10  |-  ( ( ( ( ( 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 4871 . . . . . . . . . 10  |-  ( ( q  e.  X  /\  p  e.  X )  -> 
<. q ,  p >.  e.  ( X  X.  X
) )
2422, 23syl 17 . . . . . . . . 9  |-  ( ( ( ( ( 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 5745 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
267, 18, 253syl 18 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) )
2724, 26eleqtrrd 2552 . . . . . . . 8  |-  ( ( ( ( ( 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 )
28 fvimacnv 6012 . . . . . . . 8  |-  ( ( Fun  D  /\  <. q ,  p >.  e.  dom  D )  ->  ( ( D `  <. q ,  p >. )  e.  ( 0 [,) a )  <->  <. q ,  p >.  e.  ( `' D "
( 0 [,) a
) ) ) )
2920, 27, 28syl2anc 673 . . . . . . 7  |-  ( ( ( ( ( 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 ) ) ) )
30 opelxpi 4871 . . . . . . . . . 10  |-  ( ( p  e.  X  /\  q  e.  X )  -> 
<. p ,  q >.  e.  ( X  X.  X
) )
3121, 30syl 17 . . . . . . . . 9  |-  ( ( ( ( ( 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
) )
3231, 26eleqtrrd 2552 . . . . . . . 8  |-  ( ( ( ( ( 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 )
33 fvimacnv 6012 . . . . . . . 8  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
3420, 32, 33syl2anc 673 . . . . . . 7  |-  ( ( ( ( ( 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 ) ) ) )
3517, 29, 343bitr3d 291 . . . . . 6  |-  ( ( ( ( ( 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 ) ) ) )
36 simpr 468 . . . . . . 7  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  A  =  ( `' D " ( 0 [,) a ) ) )
3736eleq2d 2534 . . . . . 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  <->  <. q ,  p >.  e.  ( `' D " ( 0 [,) a ) ) ) )
3836eleq2d 2534 . . . . . 6  |-  ( ( ( ( ( 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 ) ) ) )
3935, 37, 383bitr4d 293 . . . . 5  |-  ( ( ( ( ( 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
) )
40 eqid 2471 . . . . . . . . 9  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
4140elrnmpt 5087 . . . . . . . 8  |-  ( 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 ) ) ) )
4241ibi 249 . . . . . . 7  |-  ( A  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4342, 1eleq2s 2567 . . . . . 6  |-  ( A  e.  F  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4443ad2antlr 741 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4539, 44r19.29a 2918 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  ( <. q ,  p >.  e.  A  <->  <.
p ,  q >.  e.  A ) )
46 df-br 4396 . . . . 5  |-  ( p `' A q  <->  <. p ,  q >.  e.  `' A )
47 vex 3034 . . . . . 6  |-  p  e. 
_V
48 vex 3034 . . . . . 6  |-  q  e. 
_V
4947, 48opelcnv 5021 . . . . 5  |-  ( <.
p ,  q >.  e.  `' A  <->  <. q ,  p >.  e.  A )
5046, 49bitri 257 . . . 4  |-  ( p `' A q  <->  <. q ,  p >.  e.  A
)
51 df-br 4396 . . . 4  |-  ( p A q  <->  <. p ,  q >.  e.  A
)
5245, 50, 513bitr4g 296 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  ( p `' A q  <->  p A
q ) )
53523impb 1227 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  p  e.  X  /\  q  e.  X )  ->  (
p `' A q  <-> 
p A q ) )
546, 2, 53eqbrrdva 5009 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  `' A  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757    C_ wss 3390   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   RR*cxr 9692   RR+crp 11325   [,)cico 11662  PsMetcpsmet 19031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-xadd 11433  df-psmet 19039
This theorem is referenced by:  metust  21651
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