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Theorem iccntr 18724
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 9064 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 9064 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
3 icc0 10897 . . . . . . . 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 5673 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  =  ( ( int `  ( topGen `  ran  (,) )
) `  (/) ) )
7 retop 18667 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  Top
8 ntr0 17069 . . . . . . 7  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (/) )  =  (/) )
97, 8ax-mp 8 . . . . . 6  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  (/) )  =  (/)
10 0ss 3600 . . . . . 6  |-  (/)  C_  ( { A ,  B }  u.  ( A (,) B
) )
119, 10eqsstri 3322 . . . . 5  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  (/) )  C_  ( { A ,  B }  u.  ( A (,) B
) )
126, 11syl6eqss 3342 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( { A ,  B }  u.  ( A (,) B ) ) )
13 iccssre 10925 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
14 uniretop 18668 . . . . . . . 8  |-  RR  =  U. ( topGen `  ran  (,) )
1514ntrss2 17045 . . . . . . 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 3435 . . . . . . . 8  |-  ( { A ,  B }  u.  ( A (,) B
) )  =  ( ( A (,) B
)  u.  { A ,  B } )
20 prunioo 10958 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A (,) B
)  u.  { A ,  B } )  =  ( A [,] B
) )
2119, 20syl5eq 2432 . . . . . . 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 3329 . . . 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 9115 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  C_  ( { A ,  B }  u.  ( A (,) B ) ) )
2814ntropn 17037 . . . . . . . . 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 2388 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
3130rexmet 18694 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
32 eqid 2388 . . . . . . . . . . 11  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
3330, 32tgioo 18699 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
3433mopni2 18414 . . . . . . . . 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 10569 . . . . . . . . . . . 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 10609 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  <  A
)
4139rpred 10581 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR )
4237, 41resubcld 9398 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  e.  RR )
4342, 37ltnled 9153 . . . . . . . . . 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 10551 . . . . . . . . . . . . . . 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 10571 . . . . . . . . . . . . . . 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 9572 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  <  ( A  -  ( x  /  2 ) ) )
5037, 46readdcld 9049 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  +  x )  e.  RR )
51 ltaddrp 10577 . . . . . . . . . . . . . . 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 9164 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  <  ( A  +  x )
)
5437, 46resubcld 9398 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  e.  RR )
5554rexrd 9068 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  e.  RR* )
5650rexrd 9068 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  +  x )  e.  RR* )
57 elioo2 10890 . . . . . . . . . . . . . 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 18695 . . . . . . . . . . . . 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 2465 . . . . . . . . . . 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 3286 . . . . . . . . . . 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 3291 . . . . . . . . . 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 10908 . . . . . . . . . . . 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 2753 . . . . . . 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 18414 . . . . . . . . 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 10610 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  B  <  ( B  +  ( x  /  2 ) ) )
8179rpred 10581 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR )
8278, 81readdcld 9049 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  e.  RR )
8378, 82ltnled 9153 . . . . . . . . . 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 9398 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  e.  RR )
87 ltsubrp 10576 . . . . . . . . . . . . . . 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 9164 . . . . . . . . . . . . 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 9162 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  <  ( B  +  x )
)
9286rexrd 9068 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  e.  RR* )
9378, 85readdcld 9049 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  x )  e.  RR )
9493rexrd 9068 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  x )  e.  RR* )
95 elioo2 10890 . . . . . . . . . . . . . 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 18695 . . . . . . . . . . . . 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 2465 . . . . . . . . . . 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 3286 . . . . . . . . . . 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 3291 . . . . . . . . . 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 10908 . . . . . . . . . . . 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 2753 . . . . . . 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 2448 . . . . . . . 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 2448 . . . . . . . 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 3801 . . . . . 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 3613 . . . . 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 3629 . . . 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 18672 . . . 4  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
125 ioossicc 10929 . . . 4  |-  ( A (,) B )  C_  ( A [,] B )
12614ssntr 17046 . . . 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 3309 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 1649    e. wcel 1717   A.wral 2650   E.wrex 2651    u. cun 3262    i^i cin 3263    C_ wss 3264   (/)c0 3572   {cpr 3759   class class class wbr 4154    X. cxp 4817   ran crn 4820    |` cres 4821    o. ccom 4823   ` cfv 5395  (class class class)co 6021   RRcr 8923    + caddc 8927   RR*cxr 9053    < clt 9054    <_ cle 9055    - cmin 9224    / cdiv 9610   2c2 9982   RR+crp 10545   (,)cioo 10849   [,]cicc 10852   abscabs 11967   topGenctg 13593   * Metcxmt 16613   ballcbl 16615   MetOpencmopn 16618   Topctop 16882   intcnt 17005
This theorem is referenced by:  rolle  19742  cmvth  19743  mvth  19744  dvlip  19745  dvlipcn  19746  dvlip2  19747  c1liplem1  19748  dvgt0lem1  19754  dvle  19759  lhop1lem  19765  dvcnvrelem1  19769  dvcvx  19772  dvfsumabs  19775  ftc1cn  19795  ftc2  19796  ftc2ditglem  19797  itgparts  19799  itgsubstlem  19800  efcvx  20233  pige3  20293  logccv  20422  lgamgulmlem2  24594  ftc1cnnc  25980  areacirc  25989  lhe4.4ex1a  27216  itgsin0pilem1  27413  itgsinexplem1  27417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ico 10855  df-icc 10856  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-topgen 13595  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-top 16887  df-bases 16889  df-topon 16890  df-ntr 17008
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