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Theorem metustsymOLD 20148
Description: Elements of the filter base generated by the metric  D are symmetric. (Contributed by Thierry Arnoux, 28-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
metustsymOLD  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  `' A  =  A )
Distinct variable groups:    D, a    X, a    A, a    F, a

Proof of Theorem metustsymOLD
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
) ) )
21metustssOLD 20140 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  C_  ( X  X.  X ) )
3 cnvss 5024 . . . 4  |-  ( A 
C_  ( X  X.  X )  ->  `' A  C_  `' ( X  X.  X ) )
4 cnvxp 5267 . . . 4  |-  `' ( X  X.  X )  =  ( X  X.  X )
53, 4syl6sseq 3414 . . 3  |-  ( A 
C_  ( X  X.  X )  ->  `' A  C_  ( X  X.  X ) )
62, 5syl 16 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  `' A  C_  ( X  X.  X
) )
7 simp-4l 765 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  D  e.  ( *Met `  X ) )
8 simpr1r 1046 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  (
( p  e.  X  /\  q  e.  X
)  /\  a  e.  RR+ 
/\  A  =  ( `' D " ( 0 [,) a ) ) ) )  ->  q  e.  X )
983anassrs 1209 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
q  e.  X )
10 simpr1l 1045 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  (
( p  e.  X  /\  q  e.  X
)  /\  a  e.  RR+ 
/\  A  =  ( `' D " ( 0 [,) a ) ) ) )  ->  p  e.  X )
11103anassrs 1209 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  p  e.  X )
12 xmetsym 19934 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  q  e.  X  /\  p  e.  X
)  ->  ( q D p )  =  ( p D q ) )
137, 9, 11, 12syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( *Met `  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 6106 . . . . . . . . 9  |-  ( q D p )  =  ( D `  <. q ,  p >. )
15 df-ov 6106 . . . . . . . . 9  |-  ( p D q )  =  ( D `  <. p ,  q >. )
1613, 14, 153eqtr3g 2498 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  -> 
( D `  <. q ,  p >. )  =  ( D `  <. p ,  q >.
) )
1716eleq1d 2509 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( *Met `  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 xmetf 19916 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
19 ffun 5573 . . . . . . . . 9  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
207, 18, 193syl 20 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  Fun  D )
21 simpllr 758 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( *Met `  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 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( *Met `  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 4883 . . . . . . . . . 10  |-  ( ( q  e.  X  /\  p  e.  X )  -> 
<. q ,  p >.  e.  ( X  X.  X
) )
2422, 23syl 16 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( *Met `  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 5575 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
267, 18, 253syl 20 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  dom  D  =  ( X  X.  X ) )
2724, 26eleqtrrd 2520 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. q ,  p >.  e. 
dom  D )
28 fvimacnv 5830 . . . . . . . 8  |-  ( ( Fun  D  /\  <. q ,  p >.  e.  dom  D )  ->  ( ( D `  <. q ,  p >. )  e.  ( 0 [,) a )  <->  <. q ,  p >.  e.  ( `' D "
( 0 [,) a
) ) ) )
2920, 27, 28syl2anc 661 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( *Met `  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 4883 . . . . . . . . . 10  |-  ( ( p  e.  X  /\  q  e.  X )  -> 
<. p ,  q >.  e.  ( X  X.  X
) )
3121, 30syl 16 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( *Met `  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 2520 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. p ,  q >.  e.  dom  D )
33 fvimacnv 5830 . . . . . . . 8  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
3420, 32, 33syl2anc 661 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( *Met `  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 283 . . . . . 6  |-  ( ( ( ( ( D  e.  ( *Met `  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 461 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  A  =  ( `' D " ( 0 [,) a ) ) )
3736eleq2d 2510 . . . . . 6  |-  ( ( ( ( ( D  e.  ( *Met `  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 2510 . . . . . 6  |-  ( ( ( ( ( D  e.  ( *Met `  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 285 . . . . 5  |-  ( ( ( ( ( D  e.  ( *Met `  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 2443 . . . . . . . . 9  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
4140elrnmpt 5098 . . . . . . . 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 241 . . . . . . 7  |-  ( A  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4342, 1eleq2s 2535 . . . . . 6  |-  ( A  e.  F  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4443ad2antlr 726 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
4539, 44r19.29a 2874 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  ( <. q ,  p >.  e.  A  <->  <.
p ,  q >.  e.  A ) )
46 df-br 4305 . . . . 5  |-  ( p `' A q  <->  <. p ,  q >.  e.  `' A )
47 vex 2987 . . . . . 6  |-  p  e. 
_V
48 vex 2987 . . . . . 6  |-  q  e. 
_V
4947, 48opelcnv 5033 . . . . 5  |-  ( <.
p ,  q >.  e.  `' A  <->  <. q ,  p >.  e.  A )
5046, 49bitri 249 . . . 4  |-  ( p `' A q  <->  <. q ,  p >.  e.  A
)
51 df-br 4305 . . . 4  |-  ( p A q  <->  <. p ,  q >.  e.  A
)
5245, 50, 513bitr4g 288 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  ( p `' A q  <->  p A
q ) )
53523impb 1183 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  p  e.  X  /\  q  e.  X )  ->  (
p `' A q  <-> 
p A q ) )
546, 2, 53eqbrrdva 5021 1  |-  ( ( D  e.  ( *Met `  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 2728    C_ wss 3340   <.cop 3895   class class class wbr 4304    e. cmpt 4362    X. cxp 4850   `'ccnv 4851   dom cdm 4852   ran crn 4853   "cima 4855   Fun wfun 5424   -->wf 5426   ` cfv 5430  (class class class)co 6103   0cc0 9294   RR*cxr 9429   RR+crp 11003   [,)cico 11314   *Metcxmt 17813
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-xadd 11102  df-xmet 17822
This theorem is referenced by:  metustOLD  20154
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