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Theorem metuel2 21566
Description: Elementhood in the uniform structure generated by a metric  D (Contributed by Thierry Arnoux, 24-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metuel2.u  |-  U  =  (metUnif `  D )
Assertion
Ref Expression
metuel2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  U  <->  ( V  C_  ( X  X.  X
)  /\  E. d  e.  RR+  A. x  e.  X  A. y  e.  X  ( ( x D y )  < 
d  ->  x V
y ) ) ) )
Distinct variable groups:    x, d,
y, D    V, d, x, y    X, d, x, y
Allowed substitution hints:    U( x, y, d)

Proof of Theorem metuel2
Dummy variables  a  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metuel2.u . . . 4  |-  U  =  (metUnif `  D )
21eleq2i 2500 . . 3  |-  ( V  e.  U  <->  V  e.  (metUnif `  D ) )
32a1i 11 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  U  <->  V  e.  (metUnif `  D ) ) )
4 metuel 21565 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  (metUnif `  D )  <->  ( V  C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V
) ) )
5 vex 3084 . . . . . . . . . . 11  |-  w  e. 
_V
6 oveq2 6309 . . . . . . . . . . . . . 14  |-  ( a  =  d  ->  (
0 [,) a )  =  ( 0 [,) d ) )
76imaeq2d 5183 . . . . . . . . . . . . 13  |-  ( a  =  d  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) d
) ) )
87cbvmptv 4513 . . . . . . . . . . . 12  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
98elrnmpt 5096 . . . . . . . . . . 11  |-  ( w  e.  _V  ->  (
w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. d  e.  RR+  w  =  ( `' D " ( 0 [,) d ) ) ) )
105, 9ax-mp 5 . . . . . . . . . 10  |-  ( w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. d  e.  RR+  w  =  ( `' D " ( 0 [,) d ) ) )
1110anbi1i 699 . . . . . . . . 9  |-  ( ( w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  /\  w  C_  V )  <->  ( E. d  e.  RR+  w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V ) )
12 r19.41v 2980 . . . . . . . . 9  |-  ( E. d  e.  RR+  (
w  =  ( `' D " ( 0 [,) d ) )  /\  w  C_  V
)  <->  ( E. d  e.  RR+  w  =  ( `' D " ( 0 [,) d ) )  /\  w  C_  V
) )
1311, 12bitr4i 255 . . . . . . . 8  |-  ( ( w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  /\  w  C_  V )  <->  E. d  e.  RR+  ( w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V ) )
1413exbii 1712 . . . . . . 7  |-  ( E. w ( w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )  /\  w  C_  V )  <->  E. w E. d  e.  RR+  (
w  =  ( `' D " ( 0 [,) d ) )  /\  w  C_  V
) )
15 df-rex 2781 . . . . . . 7  |-  ( E. w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V  <->  E. w ( w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )  /\  w  C_  V ) )
16 rexcom4 3101 . . . . . . 7  |-  ( E. d  e.  RR+  E. w
( w  =  ( `' D " ( 0 [,) d ) )  /\  w  C_  V
)  <->  E. w E. d  e.  RR+  ( w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V ) )
1714, 15, 163bitr4i 280 . . . . . 6  |-  ( E. w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V  <->  E. d  e.  RR+  E. w
( w  =  ( `' D " ( 0 [,) d ) )  /\  w  C_  V
) )
18 cnvexg 6749 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
19 imaexg 6740 . . . . . . . . 9  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
20 sseq1 3485 . . . . . . . . . 10  |-  ( w  =  ( `' D " ( 0 [,) d
) )  ->  (
w  C_  V  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2120ceqsexgv 3204 . . . . . . . . 9  |-  ( ( `' D " ( 0 [,) d ) )  e.  _V  ->  ( E. w ( w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V )  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2218, 19, 213syl 18 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  ( E. w ( w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V )  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2322rexbidv 2939 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  ( E. d  e.  RR+  E. w
( w  =  ( `' D " ( 0 [,) d ) )  /\  w  C_  V
)  <->  E. d  e.  RR+  ( `' D " ( 0 [,) d ) ) 
C_  V ) )
2423adantr 466 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  ->  ( E. d  e.  RR+  E. w
( w  =  ( `' D " ( 0 [,) d ) )  /\  w  C_  V
)  <->  E. d  e.  RR+  ( `' D " ( 0 [,) d ) ) 
C_  V ) )
2517, 24syl5bb 260 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  ->  ( E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V  <->  E. d  e.  RR+  ( `' D " ( 0 [,) d ) ) 
C_  V ) )
26 cnvimass 5203 . . . . . . . . 9  |-  ( `' D " ( 0 [,) d ) ) 
C_  dom  D
27 simpll 758 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  D  e.  (PsMet `  X )
)
28 psmetf 21308 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
29 fdm 5746 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
3027, 28, 293syl 18 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  dom  D  =  ( X  X.  X ) )
3126, 30syl5sseq 3512 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  ( `' D " ( 0 [,) d ) ) 
C_  ( X  X.  X ) )
32 ssrel2 4940 . . . . . . . 8  |-  ( ( `' D " ( 0 [,) d ) ) 
C_  ( X  X.  X )  ->  (
( `' D "
( 0 [,) d
) )  C_  V  <->  A. x  e.  X  A. y  e.  X  ( <. x ,  y >.  e.  ( `' D "
( 0 [,) d
) )  ->  <. x ,  y >.  e.  V
) ) )
3331, 32syl 17 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  (
( `' D "
( 0 [,) d
) )  C_  V  <->  A. x  e.  X  A. y  e.  X  ( <. x ,  y >.  e.  ( `' D "
( 0 [,) d
) )  ->  <. x ,  y >.  e.  V
) ) )
34 simplr 760 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  x  e.  X )
35 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  y  e.  X )
36 opelxp 4879 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  ( X  X.  X
)  <->  ( x  e.  X  /\  y  e.  X ) )
3734, 35, 36sylanbrc 668 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  <. x ,  y >.  e.  ( X  X.  X ) )
3837biantrurd 510 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
( D `  <. x ,  y >. )  e.  ( 0 [,) d
)  <->  ( <. x ,  y >.  e.  ( X  X.  X )  /\  ( D `  <. x ,  y >.
)  e.  ( 0 [,) d ) ) ) )
39 simp-4l 774 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  D  e.  (PsMet `  X )
)
40 psmetcl 21309 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  e.  RR* )
4139, 34, 35, 40syl3anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
x D y )  e.  RR* )
42413biant1d 1373 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
( 0  <_  (
x D y )  /\  ( x D y )  <  d
)  <->  ( ( x D y )  e. 
RR*  /\  0  <_  ( x D y )  /\  ( x D y )  <  d
) ) )
43 psmetge0 21314 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  0  <_  ( x D y ) )
4443biantrurd 510 . . . . . . . . . . . . . 14  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
( x D y )  <  d  <->  ( 0  <_  ( x D y )  /\  (
x D y )  <  d ) ) )
4539, 34, 35, 44syl3anc 1264 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
( x D y )  <  d  <->  ( 0  <_  ( x D y )  /\  (
x D y )  <  d ) ) )
46 0xr 9687 . . . . . . . . . . . . . 14  |-  0  e.  RR*
47 simpllr 767 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  d  e.  RR+ )
4847rpxrd 11342 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  d  e.  RR* )
49 elico1 11679 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR*  /\  d  e.  RR* )  ->  (
( x D y )  e.  ( 0 [,) d )  <->  ( (
x D y )  e.  RR*  /\  0  <_  ( x D y )  /\  ( x D y )  < 
d ) ) )
5046, 48, 49sylancr 667 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
( x D y )  e.  ( 0 [,) d )  <->  ( (
x D y )  e.  RR*  /\  0  <_  ( x D y )  /\  ( x D y )  < 
d ) ) )
5142, 45, 503bitr4d 288 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
( x D y )  <  d  <->  ( x D y )  e.  ( 0 [,) d
) ) )
52 df-ov 6304 . . . . . . . . . . . . 13  |-  ( x D y )  =  ( D `  <. x ,  y >. )
5352eleq1i 2499 . . . . . . . . . . . 12  |-  ( ( x D y )  e.  ( 0 [,) d )  <->  ( D `  <. x ,  y
>. )  e.  (
0 [,) d ) )
5451, 53syl6bb 264 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
( x D y )  <  d  <->  ( D `  <. x ,  y
>. )  e.  (
0 [,) d ) ) )
55 ffn 5742 . . . . . . . . . . . 12  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
56 elpreima 6013 . . . . . . . . . . . 12  |-  ( D  Fn  ( X  X.  X )  ->  ( <. x ,  y >.  e.  ( `' D "
( 0 [,) d
) )  <->  ( <. x ,  y >.  e.  ( X  X.  X )  /\  ( D `  <. x ,  y >.
)  e.  ( 0 [,) d ) ) ) )
5739, 28, 55, 564syl 19 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  ( <. x ,  y >.  e.  ( `' D "
( 0 [,) d
) )  <->  ( <. x ,  y >.  e.  ( X  X.  X )  /\  ( D `  <. x ,  y >.
)  e.  ( 0 [,) d ) ) ) )
5838, 54, 573bitr4d 288 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
( x D y )  <  d  <->  <. x ,  y >.  e.  ( `' D " ( 0 [,) d ) ) ) )
5958anasss 651 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( x D y )  <  d  <->  <.
x ,  y >.  e.  ( `' D "
( 0 [,) d
) ) ) )
60 df-br 4421 . . . . . . . . . 10  |-  ( x V y  <->  <. x ,  y >.  e.  V
)
6160a1i 11 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x V y  <->  <. x ,  y >.  e.  V ) )
6259, 61imbi12d 321 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( ( x D y )  < 
d  ->  x V
y )  <->  ( <. x ,  y >.  e.  ( `' D " ( 0 [,) d ) )  ->  <. x ,  y
>.  e.  V ) ) )
63622ralbidva 2867 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  ( A. x  e.  X  A. y  e.  X  ( ( x D y )  <  d  ->  x V y )  <->  A. x  e.  X  A. y  e.  X  ( <. x ,  y
>.  e.  ( `' D " ( 0 [,) d
) )  ->  <. x ,  y >.  e.  V
) ) )
6433, 63bitr4d 259 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  (
( `' D "
( 0 [,) d
) )  C_  V  <->  A. x  e.  X  A. y  e.  X  (
( x D y )  <  d  ->  x V y ) ) )
6564rexbidva 2936 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  ->  ( E. d  e.  RR+  ( `' D " ( 0 [,) d ) ) 
C_  V  <->  E. d  e.  RR+  A. x  e.  X  A. y  e.  X  ( ( x D y )  < 
d  ->  x V
y ) ) )
6625, 65bitrd 256 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  ->  ( E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V  <->  E. d  e.  RR+  A. x  e.  X  A. y  e.  X  ( (
x D y )  <  d  ->  x V y ) ) )
6766pm5.32da 645 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ( V  C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V
)  <->  ( V  C_  ( X  X.  X
)  /\  E. d  e.  RR+  A. x  e.  X  A. y  e.  X  ( ( x D y )  < 
d  ->  x V
y ) ) ) )
6867adantl 467 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( ( V  C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V
)  <->  ( V  C_  ( X  X.  X
)  /\  E. d  e.  RR+  A. x  e.  X  A. y  e.  X  ( ( x D y )  < 
d  ->  x V
y ) ) ) )
693, 4, 683bitrd 282 1  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  U  <->  ( V  C_  ( X  X.  X
)  /\  E. d  e.  RR+  A. x  e.  X  A. y  e.  X  ( ( x D y )  < 
d  ->  x V
y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081    C_ wss 3436   (/)c0 3761   <.cop 4002   class class class wbr 4420    |-> cmpt 4479    X. cxp 4847   `'ccnv 4848   dom cdm 4849   ran crn 4850   "cima 4852    Fn wfn 5592   -->wf 5593   ` cfv 5597  (class class class)co 6301   0cc0 9539   RR*cxr 9674    < clt 9675    <_ cle 9676   RR+crp 11302   [,)cico 11637  PsMetcpsmet 18941  metUnifcmetu 18948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-po 4770  df-so 4771  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-1st 6803  df-2nd 6804  df-er 7367  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-2 10668  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-psmet 18949  df-fbas 18954  df-fg 18955  df-metu 18956
This theorem is referenced by: (None)
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