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Theorem iccntr 18342
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 8893 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
2 rexr 8893 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
3 icc0 10720 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  =  (/)  <->  B  <  A ) )
41, 2, 3syl2an 463 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A [,] B )  =  (/)  <->  B  <  A ) )
54biimpar 471 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( A [,] B )  =  (/) )
65fveq2d 5545 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  =  ( ( int `  ( topGen `  ran  (,) )
) `  (/) ) )
7 retop 18286 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  Top
8 ntr0 16834 . . . . . . . 8  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (/) )  =  (/) )
97, 8ax-mp 8 . . . . . . 7  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  (/) )  =  (/)
10 0ss 3496 . . . . . . 7  |-  (/)  C_  ( { A ,  B }  u.  ( A (,) B
) )
119, 10eqsstri 3221 . . . . . 6  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  (/) )  C_  ( { A ,  B }  u.  ( A (,) B
) )
1211a1i 10 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (/) )  C_  ( { A ,  B }  u.  ( A (,) B ) ) )
136, 12eqsstrd 3225 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( { A ,  B }  u.  ( A (,) B ) ) )
14 iccssre 10747 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
15 uniretop 18287 . . . . . . . 8  |-  RR  =  U. ( topGen `  ran  (,) )
1615ntrss2 16810 . . . . . . 7  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  C_  RR )  ->  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) )  C_  ( A [,] B ) )
177, 14, 16sylancr 644 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  C_  ( A [,] B ) )
1817adantr 451 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( A [,] B ) )
191, 2anim12i 549 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  RR*  /\  B  e.  RR* )
)
20 uncom 3332 . . . . . . . 8  |-  ( { A ,  B }  u.  ( A (,) B
) )  =  ( ( A (,) B
)  u.  { A ,  B } )
21 prunioo 10780 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A (,) B
)  u.  { A ,  B } )  =  ( A [,] B
) )
2220, 21syl5eq 2340 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( { A ,  B }  u.  ( A (,) B
) )  =  ( A [,] B ) )
23223expa 1151 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  ( { A ,  B }  u.  ( A (,) B ) )  =  ( A [,] B ) )
2419, 23sylan 457 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( { A ,  B }  u.  ( A (,) B
) )  =  ( A [,] B ) )
2518, 24sseqtr4d 3228 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) 
C_  ( { A ,  B }  u.  ( A (,) B ) ) )
26 simpr 447 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
27 simpl 443 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
2813, 25, 26, 27ltlecasei 8944 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  C_  ( { A ,  B }  u.  ( A (,) B ) ) )
2915ntropn 16802 . . . . . . . . 9  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  C_  RR )  ->  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) )  e.  ( topGen ` 
ran  (,) ) )
307, 14, 29sylancr 644 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  e.  ( topGen `  ran  (,) )
)
31 eqid 2296 . . . . . . . . . 10  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
3231rexmet 18313 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
33 eqid 2296 . . . . . . . . . . 11  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
3431, 33tgioo 18318 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
3534mopni2 18055 . . . . . . . . 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 ) ) )
3632, 35mp3an1 1264 . . . . . . . 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 ) ) )
3730, 36sylan 457 . . . . . . 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 ) ) )
3827ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  A  e.  RR )
39 rphalfcl 10394 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  /  2 )  e.  RR+ )
4039adantl 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR+ )
4138, 40ltsubrpd 10434 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  <  A
)
4240rpred 10406 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR )
4338, 42resubcld 9227 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  e.  RR )
4443, 38ltnled 8982 . . . . . . . . . 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 ) ) ) )
4541, 44mpbid 201 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  -.  A  <_  ( A  -  ( x  /  2 ) ) )
46 rpre 10376 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  x  e.  RR )
4746adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  x  e.  RR )
48 rphalflt 10396 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  ( x  /  2 )  < 
x )
4948adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  <  x
)
5042, 47, 38, 49ltsub2dd 9401 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  <  ( A  -  ( x  /  2 ) ) )
5138, 47readdcld 8878 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  +  x )  e.  RR )
52 ltaddrp 10402 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  x  e.  RR+ )  ->  A  <  ( A  +  x ) )
5338, 52sylancom 648 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  A  <  ( A  +  x )
)
5443, 38, 51, 41, 53lttrd 8993 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  ( x  /  2
) )  <  ( A  +  x )
)
5538, 47resubcld 9227 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  e.  RR )
5655rexrd 8897 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  -  x )  e.  RR* )
5751rexrd 8897 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( A  +  x )  e.  RR* )
58 elioo2 10713 . . . . . . . . . . . . . 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
) ) ) )
5956, 57, 58syl2anc 642 . . . . . . . . . . . . 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
) ) ) )
6043, 50, 54, 59mpbir3and 1135 . . . . . . . . . . . 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 ) ) )
6131bl2ioo 18314 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) x )  =  ( ( A  -  x ) (,) ( A  +  x )
) )
6238, 47, 61syl2anc 642 . . . . . . . . . . . 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 ) ) )
6360, 62eleqtrrd 2373 . . . . . . . . . . 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 ) )
64 ssel 3187 . . . . . . . . . . 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 ) ) ) )
6563, 64syl5com 26 . . . . . . . . . 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 ) ) ) )
6617ad2antrr 706 . . . . . . . . . . 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 ) )
6766sseld 3192 . . . . . . . . . 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 ) ) )
68 elicc2 10731 . . . . . . . . . . . 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 )
) )
69 simp2 956 . . . . . . . . . . . 12  |-  ( ( ( A  -  (
x  /  2 ) )  e.  RR  /\  A  <_  ( A  -  ( x  /  2
) )  /\  ( A  -  ( x  /  2 ) )  <_  B )  ->  A  <_  ( A  -  ( x  /  2
) ) )
7068, 69syl6bi 219 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  -  ( x  /  2
) )  e.  ( A [,] B )  ->  A  <_  ( A  -  ( x  /  2 ) ) ) )
7170ad2antrr 706 . . . . . . . . . 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 ) ) ) )
7265, 67, 713syld 51 . . . . . . . . 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 ) ) ) )
7345, 72mtod 168 . . . . . . . 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 ) ) )
7473nrexdv 2659 . . . . . . 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 ) ) )
7537, 74pm2.65da 559 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
7634mopni2 18055 . . . . . . . . 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 ) ) )
7732, 76mp3an1 1264 . . . . . . . 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 ) ) )
7830, 77sylan 457 . . . . . . 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 ) ) )
7926ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  B  e.  RR )
8039adantl 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR+ )
8179, 80ltaddrpd 10435 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  B  <  ( B  +  ( x  /  2 ) ) )
8280rpred 10406 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  e.  RR )
8379, 82readdcld 8878 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  e.  RR )
8479, 83ltnled 8982 . . . . . . . . . 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 ) )
8581, 84mpbid 201 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  -.  ( B  +  ( x  / 
2 ) )  <_  B )
8646adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  x  e.  RR )
8779, 86resubcld 9227 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  e.  RR )
88 ltsubrp 10401 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  RR  /\  x  e.  RR+ )  -> 
( B  -  x
)  <  B )
8979, 88sylancom 648 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  <  B
)
9087, 79, 83, 89, 81lttrd 8993 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  <  ( B  +  ( x  /  2 ) ) )
9148adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( x  / 
2 )  <  x
)
9282, 86, 79, 91ltadd2dd 8991 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  ( x  /  2
) )  <  ( B  +  x )
)
9387rexrd 8897 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  -  x )  e.  RR* )
9479, 86readdcld 8878 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  x )  e.  RR )
9594rexrd 8897 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  /\  x  e.  RR+ )  ->  ( B  +  x )  e.  RR* )
96 elioo2 10713 . . . . . . . . . . . . . 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
) ) ) )
9793, 95, 96syl2anc 642 . . . . . . . . . . . . 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
) ) ) )
9883, 90, 92, 97mpbir3and 1135 . . . . . . . . . . . 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 ) ) )
9931bl2ioo 18314 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  x  e.  RR )  ->  ( B ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) x )  =  ( ( B  -  x ) (,) ( B  +  x )
) )
10079, 86, 99syl2anc 642 . . . . . . . . . . . 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 ) ) )
10198, 100eleqtrrd 2373 . . . . . . . . . . 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 ) )
102 ssel 3187 . . . . . . . . . . 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 ) ) ) )
103101, 102syl5com 26 . . . . . . . . . 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 ) ) ) )
10417ad2antrr 706 . . . . . . . . . . 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 ) )
105104sseld 3192 . . . . . . . . . 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 ) ) )
106 elicc2 10731 . . . . . . . . . . . 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 )
) )
107 simp3 957 . . . . . . . . . . . 12  |-  ( ( ( B  +  ( x  /  2 ) )  e.  RR  /\  A  <_  ( B  +  ( x  /  2
) )  /\  ( B  +  ( x  /  2 ) )  <_  B )  -> 
( B  +  ( x  /  2 ) )  <_  B )
108106, 107syl6bi 219 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  ( x  /  2
) )  e.  ( A [,] B )  ->  ( B  +  ( x  /  2
) )  <_  B
) )
109108ad2antrr 706 . . . . . . . . . 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 ) )
110103, 105, 1093syld 51 . . . . . . . . 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
) )
11185, 110mtod 168 . . . . . . . 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 ) ) )
112111nrexdv 2659 . . . . . . 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 ) ) )
11378, 112pm2.65da 559 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
114 eleq1 2356 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) )  <->  A  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
115114notbid 285 . . . . . . 7  |-  ( x  =  A  ->  ( -.  x  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  <->  -.  A  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
116 eleq1 2356 . . . . . . . 8  |-  ( x  =  B  ->  (
x  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A [,] B ) )  <->  B  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
117116notbid 285 . . . . . . 7  |-  ( x  =  B  ->  ( -.  x  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  <->  -.  B  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
118115, 117ralprg 3695 . . . . . 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 ) ) ) ) )
11975, 113, 118mpbir2and 888 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  { A ,  B }  -.  x  e.  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
120 disjr 3509 . . . . 5  |-  ( ( ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  i^i 
{ A ,  B } )  =  (/)  <->  A. x  e.  { A ,  B }  -.  x  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
121119, 120sylibr 203 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  i^i 
{ A ,  B } )  =  (/) )
122 disjssun 3525 . . . 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 ) ) )
123121, 122syl 15 . . 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 ) ) )
12428, 123mpbid 201 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  C_  ( A (,) B ) )
125 iooretop 18291 . . . 4  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
126 ioossicc 10751 . . . 4  |-  ( A (,) B )  C_  ( A [,] B )
12715ssntr 16811 . . . 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 ) ) )
128125, 126, 127mpanr12 666 . . 3  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  C_  RR )  ->  ( A (,) B )  C_  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )
1297, 14, 128sylancr 644 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B
)  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )
130124, 129eqssd 3209 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {cpr 3654   class class class wbr 4039    X. cxp 4703   ran crn 4706    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874   RRcr 8752    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   abscabs 11735   topGenctg 13358   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388   Topctop 16647   intcnt 16770
This theorem is referenced by:  rolle  19353  cmvth  19354  mvth  19355  dvlip  19356  dvlipcn  19357  dvlip2  19358  c1liplem1  19359  dvgt0lem1  19365  dvle  19370  lhop1lem  19376  dvcnvrelem1  19380  dvcvx  19383  dvfsumabs  19386  ftc1cn  19406  ftc2  19407  ftc2ditglem  19408  itgparts  19410  itgsubstlem  19411  efcvx  19841  pige3  19901  logccv  20026  ftc1cnnc  25025  areacirc  25034  lhe4.4ex1a  27649  itgsin0pilem1  27847  itgsinexplem1  27851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-ntr 16773
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