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Theorem metuel2 20134
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 2502 . . 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 20133 . 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 2970 . . . . . . . . . . 11  |-  w  e. 
_V
6 oveq2 6094 . . . . . . . . . . . . . 14  |-  ( a  =  d  ->  (
0 [,) a )  =  ( 0 [,) d ) )
76imaeq2d 5164 . . . . . . . . . . . . 13  |-  ( a  =  d  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) d
) ) )
87cbvmptv 4378 . . . . . . . . . . . 12  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
98elrnmpt 5081 . . . . . . . . . . 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 2868 . . . . . . . . 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 1634 . . . . . . 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 2716 . . . . . . 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 2987 . . . . . . 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 6519 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
19 imaexg 6510 . . . . . . . . . 10  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
2018, 19syl 16 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) d ) )  e. 
_V )
21 sseq1 3372 . . . . . . . . . 10  |-  ( w  =  ( `' D " ( 0 [,) d
) )  ->  (
w  C_  V  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2221ceqsexgv 3087 . . . . . . . . 9  |-  ( ( `' D " ( 0 [,) d ) )  e.  _V  ->  ( E. w ( w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V )  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2320, 22syl 16 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  ( E. w ( w  =  ( `' D "
( 0 [,) d
) )  /\  w  C_  V )  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2423rexbidv 2731 . . . . . . 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 ) )
2524adantr 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 ) )
2617, 25syl5bb 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 ) )
27 cnvimass 5184 . . . . . . . . 9  |-  ( `' D " ( 0 [,) d ) ) 
C_  dom  D
28 simpll 753 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  D  e.  (PsMet `  X )
)
29 psmetf 19862 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
30 fdm 5558 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
3128, 29, 303syl 20 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  dom  D  =  ( X  X.  X ) )
3227, 31syl5sseq 3399 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  ( `' D " ( 0 [,) d ) ) 
C_  ( X  X.  X ) )
33 ssrel2 4925 . . . . . . . 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
) ) )
3432, 33syl 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
) ) )
35 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  x  e.  X )
36 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  y  e.  X )
37 opelxp 4864 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  ( X  X.  X
)  <->  ( x  e.  X  /\  y  e.  X ) )
3835, 36, 37sylanbrc 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 ) )
3938biantrurd 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 ) ) ) )
40 simp-4l 765 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  D  e.  (PsMet `  X )
)
41 psmetcl 19863 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  e.  RR* )
4240, 35, 36, 41syl3anc 1218 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  (
x D y )  e.  RR* )
4342biantrurd 508 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 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 ) ) ) )
44 3anass 969 . . . . . . . . . . . . . 14  |-  ( ( ( x D y )  e.  RR*  /\  0  <_  ( x D y )  /\  ( x D y )  < 
d )  <->  ( (
x D y )  e.  RR*  /\  (
0  <_  ( x D y )  /\  ( x D y )  <  d ) ) )
4543, 44syl6bbr 263 . . . . . . . . . . . . 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
) ) )
46 psmetge0 19868 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  0  <_  ( x D y ) )
4746biantrurd 508 . . . . . . . . . . . . . 14  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
( x D y )  <  d  <->  ( 0  <_  ( x D y )  /\  (
x D y )  <  d ) ) )
4840, 35, 36, 47syl3anc 1218 . . . . . . . . . . . . 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 ) ) )
49 0xr 9422 . . . . . . . . . . . . . . 15  |-  0  e.  RR*
5049a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  0  e.  RR* )
51 simpllr 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  d  e.  RR+ )
5251rpxrd 11020 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  d  e.  RR* )
53 elico1 11335 . . . . . . . . . . . . . 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 ) ) )
5450, 52, 53syl2anc 661 . . . . . . . . . . . . 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 ) ) )
5545, 48, 543bitr4d 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
) ) )
56 df-ov 6089 . . . . . . . . . . . . 13  |-  ( x D y )  =  ( D `  <. x ,  y >. )
5756eleq1i 2501 . . . . . . . . . . . 12  |-  ( ( x D y )  e.  ( 0 [,) d )  <->  ( D `  <. x ,  y
>. )  e.  (
0 [,) d ) )
5855, 57syl6bb 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 ) ) )
59 ffn 5554 . . . . . . . . . . . 12  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
60 elpreima 5818 . . . . . . . . . . . 12  |-  ( D  Fn  ( X  X.  X )  ->  ( <. x ,  y >.  e.  ( `' D "
( 0 [,) d
) )  <->  ( <. x ,  y >.  e.  ( X  X.  X )  /\  ( D `  <. x ,  y >.
)  e.  ( 0 [,) d ) ) ) )
6140, 29, 59, 604syl 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 ) ) ) )
6239, 58, 613bitr4d 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 ) ) ) )
6362anasss 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
) ) ) )
64 df-br 4288 . . . . . . . . . 10  |-  ( x V y  <->  <. x ,  y >.  e.  V
)
6564a1i 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 ) )
6663, 65imbi12d 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 ) ) )
67662ralbidva 2750 . . . . . . 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
) ) )
6834, 67bitr4d 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 ) ) )
6968rexbidva 2727 . . . . 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 ) ) )
7026, 69bitrd 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 ) ) )
7170pm5.32da 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 ) ) ) )
7271adantl 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 ) ) ) )
733, 4, 723bitrd 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 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711   _Vcvv 2967    C_ wss 3323   (/)c0 3632   <.cop 3878   class class class wbr 4287    e. cmpt 4345    X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   0cc0 9274   RR*cxr 9409    < clt 9410    <_ cle 9411   RR+crp 10983   [,)cico 11294  PsMetcpsmet 17780  metUnifcmetu 17788
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-2 10372  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ico 11298  df-psmet 17789  df-fbas 17794  df-fg 17795  df-metu 17797
This theorem is referenced by: (None)
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