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Theorem metuel2 20285
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 2532 . . 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 20284 . 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 3079 . . . . . . . . . . 11  |-  w  e. 
_V
6 oveq2 6207 . . . . . . . . . . . . . 14  |-  ( a  =  d  ->  (
0 [,) a )  =  ( 0 [,) d ) )
76imaeq2d 5276 . . . . . . . . . . . . 13  |-  ( a  =  d  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) d
) ) )
87cbvmptv 4490 . . . . . . . . . . . 12  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
98elrnmpt 5193 . . . . . . . . . . 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 2977 . . . . . . . . 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 1635 . . . . . . 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 2804 . . . . . . 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 3096 . . . . . . 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 6633 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
19 imaexg 6624 . . . . . . . . . 10  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
2018, 19syl 16 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) d ) )  e. 
_V )
21 sseq1 3484 . . . . . . . . . 10  |-  ( w  =  ( `' D " ( 0 [,) d
) )  ->  (
w  C_  V  <->  ( `' D " ( 0 [,) d ) )  C_  V ) )
2221ceqsexgv 3197 . . . . . . . . 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 2864 . . . . . . 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 5296 . . . . . . . . 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 20013 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
30 fdm 5670 . . . . . . . . . 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 3511 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  V  C_  ( X  X.  X
) )  /\  d  e.  RR+ )  ->  ( `' D " ( 0 [,) d ) ) 
C_  ( X  X.  X ) )
33 ssrel2 5037 . . . . . . . 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 4976 . . . . . . . . . . . . 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 20014 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  e.  RR* )
4240, 35, 36, 41syl3anc 1219 . . . . . . . . . . . . . . 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 20019 . . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 9540 . . . . . . . . . . . . . . 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 11138 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  V  C_  ( X  X.  X ) )  /\  d  e.  RR+ )  /\  x  e.  X
)  /\  y  e.  X )  ->  d  e.  RR* )
53 elico1 11453 . . . . . . . . . . . . . 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 6202 . . . . . . . . . . . . 13  |-  ( x D y )  =  ( D `  <. x ,  y >. )
5756eleq1i 2531 . . . . . . . . . . . 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 5666 . . . . . . . . . . . 12  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
60 elpreima 5931 . . . . . . . . . . . 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 4400 . . . . . . . . . 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 2849 . . . . . . 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 2861 . . . . 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 1370   E.wex 1587    e. wcel 1758    =/= wne 2647   A.wral 2798   E.wrex 2799   _Vcvv 3076    C_ wss 3435   (/)c0 3744   <.cop 3990   class class class wbr 4399    |-> cmpt 4457    X. cxp 4945   `'ccnv 4946   dom cdm 4947   ran crn 4948   "cima 4950    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199   0cc0 9392   RR*cxr 9527    < clt 9528    <_ cle 9529   RR+crp 11101   [,)cico 11412  PsMetcpsmet 17924  metUnifcmetu 17932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-2 10490  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ico 11416  df-psmet 17933  df-fbas 17938  df-fg 17939  df-metu 17941
This theorem is referenced by: (None)
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