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Theorem metuel2 21208
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 2535 . . 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 21207 . 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 3112 . . . . . . . . . . 11  |-  w  e. 
_V
6 oveq2 6304 . . . . . . . . . . . . . 14  |-  ( a  =  d  ->  (
0 [,) a )  =  ( 0 [,) d ) )
76imaeq2d 5347 . . . . . . . . . . . . 13  |-  ( a  =  d  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) d
) ) )
87cbvmptv 4548 . . . . . . . . . . . 12  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
98elrnmpt 5259 . . . . . . . . . . 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 695 . . . . . . . . 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 3009 . . . . . . . . 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 252 . . . . . . . 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 1668 . . . . . . 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 2813 . . . . . . 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 3129 . . . . . . 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 277 . . . . . 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 6745 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
19 imaexg 6736 . . . . . . . . 9  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
20 sseq1 3520 . . . . . . . . . 10  |-  ( w  =  ( `' D " ( 0 [,) d
) )  ->  (
w  C_  V  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2120ceqsexgv 3232 . . . . . . . . 9  |-  ( ( `' D " ( 0 [,) d ) )  e.  _V  ->  ( E. w ( w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V )  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2218, 19, 213syl 20 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  ( E. w ( w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V )  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2322rexbidv 2968 . . . . . . 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 465 . . . . . 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 257 . . . . 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 5367 . . . . . . . . 9  |-  ( `' D " ( 0 [,) d ) ) 
C_  dom  D
27 simpll 753 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  D  e.  (PsMet `  X )
)
28 psmetf 20936 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
29 fdm 5741 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
3027, 28, 293syl 20 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  dom  D  =  ( X  X.  X ) )
3126, 30syl5sseq 3547 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  ( `' D " ( 0 [,) d ) ) 
C_  ( X  X.  X ) )
32 ssrel2 5102 . . . . . . . 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 16 . . . . . . 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 755 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  x  e.  X )
35 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  y  e.  X )
36 opelxp 5038 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  ( X  X.  X
)  <->  ( x  e.  X  /\  y  e.  X ) )
3734, 35, 36sylanbrc 664 . . . . . . . . . . . 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 508 . . . . . . . . . . 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 767 . . . . . . . . . . . . . . 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 20937 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  e.  RR* )
4139, 34, 35, 40syl3anc 1228 . . . . . . . . . . . . . 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 1337 . . . . . . . . . . . . 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 20942 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  0  <_  ( x D y ) )
4443biantrurd 508 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 9657 . . . . . . . . . . . . . 14  |-  0  e.  RR*
47 simpllr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  d  e.  RR+ )
4847rpxrd 11282 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  d  e.  RR* )
49 elico1 11597 . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . 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 285 . . . . . . . . . . . 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 6299 . . . . . . . . . . . . 13  |-  ( x D y )  =  ( D `  <. x ,  y >. )
5352eleq1i 2534 . . . . . . . . . . . 12  |-  ( ( x D y )  e.  ( 0 [,) d )  <->  ( D `  <. x ,  y
>. )  e.  (
0 [,) d ) )
5451, 53syl6bb 261 . . . . . . . . . . 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 5737 . . . . . . . . . . . 12  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
56 elpreima 6008 . . . . . . . . . . . 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 21 . . . . . . . . . . 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 285 . . . . . . . . . 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 647 . . . . . . . . 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 4457 . . . . . . . . . 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 320 . . . . . . . 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 2899 . . . . . . 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 256 . . . . . 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 2965 . . . . 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 253 . . . 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 641 . . 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 466 . 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 279 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 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109    C_ wss 3471   (/)c0 3793   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   RR*cxr 9644    < clt 9645    <_ cle 9646   RR+crp 11245   [,)cico 11556  PsMetcpsmet 18529  metUnifcmetu 18537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-psmet 18538  df-fbas 18543  df-fg 18544  df-metu 18546
This theorem is referenced by: (None)
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