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Theorem iccntr 18805
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 9086 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 9086 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
3 icc0 10920 . . . . . . . 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 5691 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  =  ( ( int `  ( topGen `  ran  (,) )
) `  (/) ) )
7 retop 18748 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  Top
8 ntr0 17100 . . . . . . 7  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (/) )  =  (/) )
97, 8ax-mp 8 . . . . . 6  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  (/) )  =  (/)
10 0ss 3616 . . . . . 6  |-  (/)  C_  ( { A ,  B }  u.  ( A (,) B
) )
119, 10eqsstri 3338 . . . . 5  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  (/) )  C_  ( { A ,  B }  u.  ( A (,) B
) )
126, 11syl6eqss 3358 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( { A ,  B }  u.  ( A (,) B ) ) )
13 iccssre 10948 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
14 uniretop 18749 . . . . . . . 8  |-  RR  =  U. ( topGen `  ran  (,) )
1514ntrss2 17076 . . . . . . 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 3451 . . . . . . . 8  |-  ( { A ,  B }  u.  ( A (,) B
) )  =  ( ( A (,) B
)  u.  { A ,  B } )
20 prunioo 10981 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A (,) B
)  u.  { A ,  B } )  =  ( A [,] B
) )
2119, 20syl5eq 2448 . . . . . . 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 3345 . . . 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 9137 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  C_  ( { A ,  B }  u.  ( A (,) B ) ) )
2814ntropn 17068 . . . . . . . . 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 2404 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
3130rexmet 18775 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
32 eqid 2404 . . . . . . . . . . 11  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
3330, 32tgioo 18780 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
3433mopni2 18476 . . . . . . . . 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 10592 . . . . . . . . . . . 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 10632 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  <  A
)
4139rpred 10604 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR )
4237, 41resubcld 9421 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  e.  RR )
4342, 37ltnled 9176 . . . . . . . . . 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 10574 . . . . . . . . . . . . . . 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 10594 . . . . . . . . . . . . . . 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 9595 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  <  ( A  -  ( x  /  2 ) ) )
5037, 46readdcld 9071 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  +  x )  e.  RR )
51 ltaddrp 10600 . . . . . . . . . . . . . . 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 9187 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  <  ( A  +  x )
)
5437, 46resubcld 9421 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  e.  RR )
5554rexrd 9090 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  e.  RR* )
5650rexrd 9090 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  +  x )  e.  RR* )
57 elioo2 10913 . . . . . . . . . . . . . 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 18776 . . . . . . . . . . . . 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 2481 . . . . . . . . . . 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 3302 . . . . . . . . . . 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 3307 . . . . . . . . . 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 10931 . . . . . . . . . . . 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 2769 . . . . . . 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 18476 . . . . . . . . 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 10633 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  B  <  ( B  +  ( x  /  2 ) ) )
8179rpred 10604 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR )
8278, 81readdcld 9071 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  e.  RR )
8378, 82ltnled 9176 . . . . . . . . . 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 9421 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  e.  RR )
87 ltsubrp 10599 . . . . . . . . . . . . . . 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 9187 . . . . . . . . . . . . 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 9185 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  <  ( B  +  x )
)
9286rexrd 9090 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  e.  RR* )
9378, 85readdcld 9071 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  x )  e.  RR )
9493rexrd 9090 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  x )  e.  RR* )
95 elioo2 10913 . . . . . . . . . . . . . 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 18776 . . . . . . . . . . . . 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 2481 . . . . . . . . . . 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 3302 . . . . . . . . . . 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 3307 . . . . . . . . . 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 10931 . . . . . . . . . . . 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 2769 . . . . . . 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 2464 . . . . . . . 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 2464 . . . . . . . 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 3817 . . . . . 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 3629 . . . . 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 3645 . . . 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 18753 . . . 4  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
125 ioossicc 10952 . . . 4  |-  ( A (,) B )  C_  ( A [,] B )
12614ssntr 17077 . . . 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 3325 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 1721   A.wral 2666   E.wrex 2667    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {cpr 3775   class class class wbr 4172    X. cxp 4835   ran crn 4838    |` cres 4839    o. ccom 4841   ` cfv 5413  (class class class)co 6040   RRcr 8945    + caddc 8949   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   RR+crp 10568   (,)cioo 10872   [,]cicc 10875   abscabs 11994   topGenctg 13620   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646   Topctop 16913   intcnt 17036
This theorem is referenced by:  rolle  19827  cmvth  19828  mvth  19829  dvlip  19830  dvlipcn  19831  dvlip2  19832  c1liplem1  19833  dvgt0lem1  19839  dvle  19844  lhop1lem  19850  dvcnvrelem1  19854  dvcvx  19857  dvfsumabs  19860  ftc1cn  19880  ftc2  19881  ftc2ditglem  19882  itgparts  19884  itgsubstlem  19885  efcvx  20318  pige3  20378  logccv  20507  lgamgulmlem2  24767  ftc1cnnc  26178  areacirc  26187  lhe4.4ex1a  27414  itgsin0pilem1  27611  itgsinexplem1  27615
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-ntr 17039
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