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Theorem iccntr 18852
Description: The interior of a closed interval in the standard topology on  RR is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)
Assertion
Ref Expression
iccntr  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )

Proof of Theorem iccntr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rexr 9130 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 9130 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
3 icc0 10964 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  =  (/)  <->  B  <  A ) )
41, 2, 3syl2an 464 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A [,] B )  =  (/)  <->  B  <  A ) )
54biimpar 472 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( A [,] B )  =  (/) )
65fveq2d 5732 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  =  ( ( int `  ( topGen `  ran  (,) )
) `  (/) ) )
7 retop 18795 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  Top
8 ntr0 17145 . . . . . . 7  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (/) )  =  (/) )
97, 8ax-mp 8 . . . . . 6  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  (/) )  =  (/)
10 0ss 3656 . . . . . 6  |-  (/)  C_  ( { A ,  B }  u.  ( A (,) B
) )
119, 10eqsstri 3378 . . . . 5  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  (/) )  C_  ( { A ,  B }  u.  ( A (,) B
) )
126, 11syl6eqss 3398 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( { A ,  B }  u.  ( A (,) B ) ) )
13 iccssre 10992 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
14 uniretop 18796 . . . . . . . 8  |-  RR  =  U. ( topGen `  ran  (,) )
1514ntrss2 17121 . . . . . . 7  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  C_  RR )  ->  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) )  C_  ( A [,] B ) )
167, 13, 15sylancr 645 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  C_  ( A [,] B ) )
1716adantr 452 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( A [,] B ) )
181, 2anim12i 550 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  RR*  /\  B  e.  RR* )
)
19 uncom 3491 . . . . . . . 8  |-  ( { A ,  B }  u.  ( A (,) B
) )  =  ( ( A (,) B
)  u.  { A ,  B } )
20 prunioo 11025 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A (,) B
)  u.  { A ,  B } )  =  ( A [,] B
) )
2119, 20syl5eq 2480 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( { A ,  B }  u.  ( A (,) B
) )  =  ( A [,] B ) )
22213expa 1153 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  ( { A ,  B }  u.  ( A (,) B ) )  =  ( A [,] B ) )
2318, 22sylan 458 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( { A ,  B }  u.  ( A (,) B
) )  =  ( A [,] B ) )
2417, 23sseqtr4d 3385 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( { A ,  B }  u.  ( A (,) B ) ) )
25 simpr 448 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
26 simpl 444 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
2712, 24, 25, 26ltlecasei 9181 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  C_  ( { A ,  B }  u.  ( A (,) B ) ) )
2814ntropn 17113 . . . . . . . . 9  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  C_  RR )  ->  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) )  e.  ( topGen ` 
ran  (,) ) )
297, 13, 28sylancr 645 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  e.  ( topGen `  ran  (,) )
)
30 eqid 2436 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
3130rexmet 18822 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
32 eqid 2436 . . . . . . . . . . 11  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
3330, 32tgioo 18827 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
3433mopni2 18523 . . . . . . . . 9  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR )  /\  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  e.  ( topGen `  ran  (,) )  /\  A  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  ->  E. x  e.  RR+  ( A ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
3531, 34mp3an1 1266 . . . . . . . 8  |-  ( ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  e.  ( topGen `  ran  (,) )  /\  A  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  ->  E. x  e.  RR+  ( A ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
3629, 35sylan 458 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  ->  E. x  e.  RR+  ( A ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
3726ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  A  e.  RR )
38 rphalfcl 10636 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  /  2 )  e.  RR+ )
3938adantl 453 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR+ )
4037, 39ltsubrpd 10676 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  <  A
)
4139rpred 10648 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR )
4237, 41resubcld 9465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  e.  RR )
4342, 37ltnled 9220 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( A  -  ( x  / 
2 ) )  < 
A  <->  -.  A  <_  ( A  -  ( x  /  2 ) ) ) )
4440, 43mpbid 202 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  -.  A  <_  ( A  -  ( x  /  2 ) ) )
45 rpre 10618 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  x  e.  RR )
4645adantl 453 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  x  e.  RR )
47 rphalflt 10638 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  ( x  /  2 )  < 
x )
4847adantl 453 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  <  x
)
4941, 46, 37, 48ltsub2dd 9639 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  <  ( A  -  ( x  /  2 ) ) )
5037, 46readdcld 9115 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  +  x )  e.  RR )
51 ltaddrp 10644 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  x  e.  RR+ )  ->  A  <  ( A  +  x ) )
5237, 51sylancom 649 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  A  <  ( A  +  x )
)
5342, 37, 50, 40, 52lttrd 9231 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  <  ( A  +  x )
)
5437, 46resubcld 9465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  e.  RR )
5554rexrd 9134 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  e.  RR* )
5650rexrd 9134 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  +  x )  e.  RR* )
57 elioo2 10957 . . . . . . . . . . . . . 14  |-  ( ( ( A  -  x
)  e.  RR*  /\  ( A  +  x )  e.  RR* )  ->  (
( A  -  (
x  /  2 ) )  e.  ( ( A  -  x ) (,) ( A  +  x ) )  <->  ( ( A  -  ( x  /  2 ) )  e.  RR  /\  ( A  -  x )  <  ( A  -  (
x  /  2 ) )  /\  ( A  -  ( x  / 
2 ) )  < 
( A  +  x
) ) ) )
5855, 56, 57syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( A  -  ( x  / 
2 ) )  e.  ( ( A  -  x ) (,) ( A  +  x )
)  <->  ( ( A  -  ( x  / 
2 ) )  e.  RR  /\  ( A  -  x )  < 
( A  -  (
x  /  2 ) )  /\  ( A  -  ( x  / 
2 ) )  < 
( A  +  x
) ) ) )
5942, 49, 53, 58mpbir3and 1137 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  e.  ( ( A  -  x
) (,) ( A  +  x ) ) )
6030bl2ioo 18823 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) x )  =  ( ( A  -  x ) (,) ( A  +  x )
) )
6137, 46, 60syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  =  ( ( A  -  x ) (,) ( A  +  x ) ) )
6259, 61eleqtrrd 2513 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  e.  ( A ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x ) )
63 ssel 3342 . . . . . . . . . . 11  |-  ( ( A ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  ->  ( ( A  -  ( x  / 
2 ) )  e.  ( A ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) x )  -> 
( A  -  (
x  /  2 ) )  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) ) ) )
6462, 63syl5com 28 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( A ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  ->  ( A  -  ( x  /  2
) )  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
6516ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  C_  ( A [,] B ) )
6665sseld 3347 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( A  -  ( x  / 
2 ) )  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  -> 
( A  -  (
x  /  2 ) )  e.  ( A [,] B ) ) )
67 elicc2 10975 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  -  ( x  /  2
) )  e.  ( A [,] B )  <-> 
( ( A  -  ( x  /  2
) )  e.  RR  /\  A  <_  ( A  -  ( x  / 
2 ) )  /\  ( A  -  (
x  /  2 ) )  <_  B )
) )
68 simp2 958 . . . . . . . . . . . 12  |-  ( ( ( A  -  (
x  /  2 ) )  e.  RR  /\  A  <_  ( A  -  ( x  /  2
) )  /\  ( A  -  ( x  /  2 ) )  <_  B )  ->  A  <_  ( A  -  ( x  /  2
) ) )
6967, 68syl6bi 220 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  -  ( x  /  2
) )  e.  ( A [,] B )  ->  A  <_  ( A  -  ( x  /  2 ) ) ) )
7069ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( A  -  ( x  / 
2 ) )  e.  ( A [,] B
)  ->  A  <_  ( A  -  ( x  /  2 ) ) ) )
7164, 66, 703syld 53 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( A ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  ->  A  <_  ( A  -  ( x  /  2 ) ) ) )
7244, 71mtod 170 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  -.  ( A
( ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
7372nrexdv 2809 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  ->  -.  E. x  e.  RR+  ( A (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
7436, 73pm2.65da 560 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
7533mopni2 18523 . . . . . . . . 9  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR )  /\  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  e.  ( topGen `  ran  (,) )  /\  B  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  ->  E. x  e.  RR+  ( B ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
7631, 75mp3an1 1266 . . . . . . . 8  |-  ( ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  e.  ( topGen `  ran  (,) )  /\  B  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  ->  E. x  e.  RR+  ( B ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
7729, 76sylan 458 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  ->  E. x  e.  RR+  ( B ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
7825ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  B  e.  RR )
7938adantl 453 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR+ )
8078, 79ltaddrpd 10677 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  B  <  ( B  +  ( x  /  2 ) ) )
8179rpred 10648 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR )
8278, 81readdcld 9115 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  e.  RR )
8378, 82ltnled 9220 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  < 
( B  +  ( x  /  2 ) )  <->  -.  ( B  +  ( x  / 
2 ) )  <_  B ) )
8480, 83mpbid 202 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  -.  ( B  +  ( x  / 
2 ) )  <_  B )
8545adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  x  e.  RR )
8678, 85resubcld 9465 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  e.  RR )
87 ltsubrp 10643 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  RR  /\  x  e.  RR+ )  -> 
( B  -  x
)  <  B )
8878, 87sylancom 649 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  <  B
)
8986, 78, 82, 88, 80lttrd 9231 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  <  ( B  +  ( x  /  2 ) ) )
9047adantl 453 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  <  x
)
9181, 85, 78, 90ltadd2dd 9229 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  <  ( B  +  x )
)
9286rexrd 9134 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  e.  RR* )
9378, 85readdcld 9115 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  x )  e.  RR )
9493rexrd 9134 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  x )  e.  RR* )
95 elioo2 10957 . . . . . . . . . . . . . 14  |-  ( ( ( B  -  x
)  e.  RR*  /\  ( B  +  x )  e.  RR* )  ->  (
( B  +  ( x  /  2 ) )  e.  ( ( B  -  x ) (,) ( B  +  x ) )  <->  ( ( B  +  ( x  /  2 ) )  e.  RR  /\  ( B  -  x )  <  ( B  +  ( x  /  2 ) )  /\  ( B  +  ( x  / 
2 ) )  < 
( B  +  x
) ) ) )
9692, 94, 95syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( B  +  ( x  / 
2 ) )  e.  ( ( B  -  x ) (,) ( B  +  x )
)  <->  ( ( B  +  ( x  / 
2 ) )  e.  RR  /\  ( B  -  x )  < 
( B  +  ( x  /  2 ) )  /\  ( B  +  ( x  / 
2 ) )  < 
( B  +  x
) ) ) )
9782, 89, 91, 96mpbir3and 1137 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  e.  ( ( B  -  x
) (,) ( B  +  x ) ) )
9830bl2ioo 18823 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  x  e.  RR )  ->  ( B ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) x )  =  ( ( B  -  x ) (,) ( B  +  x )
) )
9978, 85, 98syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  =  ( ( B  -  x ) (,) ( B  +  x ) ) )
10097, 99eleqtrrd 2513 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  e.  ( B ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x ) )
101 ssel 3342 . . . . . . . . . . 11  |-  ( ( B ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  ->  ( ( B  +  ( x  / 
2 ) )  e.  ( B ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) x )  -> 
( B  +  ( x  /  2 ) )  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) ) ) )
102100, 101syl5com 28 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( B ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  ->  ( B  +  ( x  /  2
) )  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
10316ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  C_  ( A [,] B ) )
104103sseld 3347 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( B  +  ( x  / 
2 ) )  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  -> 
( B  +  ( x  /  2 ) )  e.  ( A [,] B ) ) )
105 elicc2 10975 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  ( x  /  2
) )  e.  ( A [,] B )  <-> 
( ( B  +  ( x  /  2
) )  e.  RR  /\  A  <_  ( B  +  ( x  / 
2 ) )  /\  ( B  +  (
x  /  2 ) )  <_  B )
) )
106 simp3 959 . . . . . . . . . . . 12  |-  ( ( ( B  +  ( x  /  2 ) )  e.  RR  /\  A  <_  ( B  +  ( x  /  2
) )  /\  ( B  +  ( x  /  2 ) )  <_  B )  -> 
( B  +  ( x  /  2 ) )  <_  B )
107105, 106syl6bi 220 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  ( x  /  2
) )  e.  ( A [,] B )  ->  ( B  +  ( x  /  2
) )  <_  B
) )
108107ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( B  +  ( x  / 
2 ) )  e.  ( A [,] B
)  ->  ( B  +  ( x  / 
2 ) )  <_  B ) )
109102, 104, 1083syld 53 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( ( B ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  ->  ( B  +  ( x  /  2
) )  <_  B
) )
11084, 109mtod 170 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  -.  ( B
( ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
111110nrexdv 2809 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  ->  -.  E. x  e.  RR+  ( B (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) x )  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
11277, 111pm2.65da 560 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
113 eleq1 2496 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) )  <->  A  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
114113notbid 286 . . . . . . 7  |-  ( x  =  A  ->  ( -.  x  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  <->  -.  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
115 eleq1 2496 . . . . . . . 8  |-  ( x  =  B  ->  (
x  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) )  <->  B  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
116115notbid 286 . . . . . . 7  |-  ( x  =  B  ->  ( -.  x  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  <->  -.  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
117114, 116ralprg 3857 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A. x  e. 
{ A ,  B }  -.  x  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  <->  ( -.  A  e.  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  /\  -.  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) ) )
11874, 112, 117mpbir2and 889 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  { A ,  B }  -.  x  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
119 disjr 3669 . . . . 5  |-  ( ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  i^i 
{ A ,  B } )  =  (/)  <->  A. x  e.  { A ,  B }  -.  x  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
120118, 119sylibr 204 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  i^i 
{ A ,  B } )  =  (/) )
121 disjssun 3685 . . . 4  |-  ( ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  i^i 
{ A ,  B } )  =  (/)  ->  ( ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  C_  ( { A ,  B }  u.  ( A (,) B ) )  <->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( A (,) B ) ) )
122120, 121syl 16 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  C_  ( { A ,  B }  u.  ( A (,) B ) )  <->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( A (,) B ) ) )
12327, 122mpbid 202 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  C_  ( A (,) B ) )
124 iooretop 18800 . . . 4  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
125 ioossicc 10996 . . . 4  |-  ( A (,) B )  C_  ( A [,] B )
12614ssntr 17122 . . . 4  |-  ( ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B ) 
C_  RR )  /\  ( ( A (,) B )  e.  (
topGen `  ran  (,) )  /\  ( A (,) B
)  C_  ( A [,] B ) ) )  ->  ( A (,) B )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
127124, 125, 126mpanr12 667 . . 3  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  C_  RR )  ->  ( A (,) B )  C_  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
1287, 13, 127sylancr 645 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B
)  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
129123, 128eqssd 3365 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   {cpr 3815   class class class wbr 4212    X. cxp 4876   ran crn 4879    |` cres 4880    o. ccom 4882   ` cfv 5454  (class class class)co 6081   RRcr 8989    + caddc 8993   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291    / cdiv 9677   2c2 10049   RR+crp 10612   (,)cioo 10916   [,]cicc 10919   abscabs 12039   topGenctg 13665   * Metcxmt 16686   ballcbl 16688   MetOpencmopn 16691   Topctop 16958   intcnt 17081
This theorem is referenced by:  rolle  19874  cmvth  19875  mvth  19876  dvlip  19877  dvlipcn  19878  dvlip2  19879  c1liplem1  19880  dvgt0lem1  19886  dvle  19891  lhop1lem  19897  dvcnvrelem1  19901  dvcvx  19904  dvfsumabs  19907  ftc1cn  19927  ftc2  19928  ftc2ditglem  19929  itgparts  19931  itgsubstlem  19932  efcvx  20365  pige3  20425  logccv  20554  lgamgulmlem2  24814  ftc1cnnc  26279  ftc2nc  26289  areacirc  26297  lhe4.4ex1a  27523  itgsin0pilem1  27720  itgsinexplem1  27724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-ntr 17084
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