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Theorem metustsym 20828
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 20820 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X
) )
3 cnvss 5175 . . . 4  |-  ( A 
C_  ( X  X.  X )  ->  `' A  C_  `' ( X  X.  X ) )
4 cnvxp 5424 . . . 4  |-  `' ( X  X.  X )  =  ( X  X.  X )
53, 4syl6sseq 3550 . . 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 1054 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 1053 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 20577 . . . . . . . . . . 11  |-  ( ( D  e.  (PsMet `  X )  /\  q  e.  X  /\  p  e.  X )  ->  (
q D p )  =  ( p D q ) )
137, 9, 11, 12syl3anc 1228 . . . . . . . . . 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 6287 . . . . . . . . . 10  |-  ( q D p )  =  ( D `  <. q ,  p >. )
15 df-ov 6287 . . . . . . . . . 10  |-  ( p D q )  =  ( D `  <. p ,  q >. )
1613, 14, 153eqtr3g 2531 . . . . . . . . 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 2536 . . . . . . . 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 20573 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
19 ffun 5733 . . . . . . . . . 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 5031 . . . . . . . . . . 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 5735 . . . . . . . . . . . 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 2558 . . . . . . . . 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 5996 . . . . . . . . 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 5031 . . . . . . . . . . 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 2558 . . . . . . . . 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 5996 . . . . . . . . 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 2537 . . . . . . 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 2537 . . . . . . 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 2467 . . . . . . . . . 10  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
4342elrnmpt 5249 . . . . . . . . 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 2575 . . . . . . 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 3003 . . . . 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 4448 . . . . . 6  |-  ( p `' A q  <->  <. p ,  q >.  e.  `' A )
49 vex 3116 . . . . . . 7  |-  p  e. 
_V
50 vex 3116 . . . . . . 7  |-  q  e. 
_V
5149, 50opelcnv 5184 . . . . . 6  |-  ( <.
p ,  q >.  e.  `' A  <->  <. q ,  p >.  e.  A )
5248, 51bitri 249 . . . . 5  |-  ( p `' A q  <->  <. q ,  p >.  e.  A
)
53 df-br 4448 . . . . 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 1194 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  p  e.  X  /\  q  e.  X )  ->  (
p `' A q  <-> 
p A q ) )
576, 2, 56eqbrrdva 5172 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 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6284   0cc0 9492   RR*cxr 9627   RR+crp 11220   [,)cico 11531  PsMetcpsmet 18201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-xadd 11319  df-psmet 18210
This theorem is referenced by:  metust  20834
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