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Theorem icoopnst 18957
Description: A half-open interval starting at  A is open in the closed interval from  A to  B. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Hypothesis
Ref Expression
icoopnst.1  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( A [,] B )  X.  ( A [,] B ) ) ) )
Assertion
Ref Expression
icoopnst  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  ->  ( A [,) C )  e.  J
) )

Proof of Theorem icoopnst
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 iooretop 18793 . . . . 5  |-  ( ( A  -  1 ) (,) C )  e.  ( topGen `  ran  (,) )
2 simp1 957 . . . . . . . . . . 11  |-  ( ( v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  e.  RR )
32a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  e.  RR ) )
4 ltm1 9843 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  ( A  -  1 )  <  A )
54adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  -  1 )  <  A )
6 peano2rem 9360 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
76adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  -  1 )  e.  RR )
8 ltletr 9159 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  -  1 )  e.  RR  /\  A  e.  RR  /\  v  e.  RR )  ->  (
( ( A  - 
1 )  <  A  /\  A  <_  v )  ->  ( A  - 
1 )  <  v
) )
983expb 1154 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  -  1 )  e.  RR  /\  ( A  e.  RR  /\  v  e.  RR ) )  ->  ( (
( A  -  1 )  <  A  /\  A  <_  v )  -> 
( A  -  1 )  <  v ) )
107, 9mpancom 651 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( ( ( A  -  1 )  < 
A  /\  A  <_  v )  ->  ( A  -  1 )  < 
v ) )
115, 10mpand 657 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  <_  v  ->  ( A  -  1 )  <  v ) )
1211impr 603 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( v  e.  RR  /\  A  <_  v )
)  ->  ( A  -  1 )  < 
v )
13123adantr3 1118 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <  C ) )  ->  ( A  - 
1 )  <  v
)
1413ex 424 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  (
( v  e.  RR  /\  A  <_  v  /\  v  <  C )  -> 
( A  -  1 )  <  v ) )
1514ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  ( A  -  1 )  <  v ) )
16 simp3 959 . . . . . . . . . . 11  |-  ( ( v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <  C )
1716a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <  C ) )
183, 15, 173jcad 1135 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  (
v  e.  RR  /\  ( A  -  1
)  <  v  /\  v  <  C ) ) )
19 simp2 958 . . . . . . . . . . 11  |-  ( ( v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  A  <_  v )
2019a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  A  <_  v ) )
21 rexr 9123 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  A  e.  RR* )
22 elioc2 10966 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
2321, 22sylan 458 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <-> 
( C  e.  RR  /\  A  <  C  /\  C  <_  B ) ) )
2423biimpa 471 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) )
25 ltletr 9159 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
( v  <  C  /\  C  <_  B )  ->  v  <  B
) )
26 ltle 9156 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( v  e.  RR  /\  B  e.  RR )  ->  ( v  <  B  ->  v  <_  B )
)
27263adant2 976 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
v  <  B  ->  v  <_  B ) )
2825, 27syld 42 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
( v  <  C  /\  C  <_  B )  ->  v  <_  B
) )
29283expa 1153 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( v  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( v  <  C  /\  C  <_  B )  ->  v  <_  B ) )
3029an31s 782 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  ->  ( ( v  <  C  /\  C  <_  B )  ->  v  <_  B ) )
3130imp 419 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  /\  (
v  <  C  /\  C  <_  B ) )  ->  v  <_  B
)
3231ancom2s 778 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  /\  ( C  <_  B  /\  v  <  C ) )  -> 
v  <_  B )
3332an4s 800 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B )  /\  (
v  e.  RR  /\  v  <  C ) )  ->  v  <_  B
)
34333adantr2 1117 . . . . . . . . . . . . . . 15  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B )  /\  (
v  e.  RR  /\  A  <_  v  /\  v  <  C ) )  -> 
v  <_  B )
3534ex 424 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B
)  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
3635anasss 629 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  C  <_  B )
)  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
37363adantr2 1117 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  A  <  C  /\  C  <_  B ) )  ->  ( ( v  e.  RR  /\  A  <_  v  /\  v  < 
C )  ->  v  <_  B ) )
3837adantll 695 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) )  ->  (
( v  e.  RR  /\  A  <_  v  /\  v  <  C )  -> 
v  <_  B )
)
3924, 38syldan 457 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
403, 20, 393jcad 1135 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  (
v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) )
4118, 40jcad 520 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  (
( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) ) )
42 simpl1 960 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  v  e.  RR )
43 simpr2 964 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  A  <_  v
)
44 simpl3 962 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  v  <  C
)
4542, 43, 443jca 1134 . . . . . . . 8  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <  C ) )
4641, 45impbid1 195 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  <->  ( (
v  e.  RR  /\  ( A  -  1
)  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) ) )
47 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  A  e.  RR )
4824simp1d 969 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  C  e.  RR )
49 rexr 9123 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  RR* )
5048, 49syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  C  e.  RR* )
51 elico2 10967 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR* )  -> 
( v  e.  ( A [,) C )  <-> 
( v  e.  RR  /\  A  <_  v  /\  v  <  C ) ) )
5247, 50, 51syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,) C
)  <->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <  C ) ) )
53 elin 3523 . . . . . . . 8  |-  ( v  e.  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B
) )  <->  ( v  e.  ( ( A  - 
1 ) (,) C
)  /\  v  e.  ( A [,] B ) ) )
54 rexr 9123 . . . . . . . . . . . 12  |-  ( ( A  -  1 )  e.  RR  ->  ( A  -  1 )  e.  RR* )
556, 54syl 16 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR* )
5655ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A  -  1 )  e. 
RR* )
57 elioo2 10950 . . . . . . . . . 10  |-  ( ( ( A  -  1 )  e.  RR*  /\  C  e.  RR* )  ->  (
v  e.  ( ( A  -  1 ) (,) C )  <->  ( v  e.  RR  /\  ( A  -  1 )  < 
v  /\  v  <  C ) ) )
5856, 50, 57syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( ( A  - 
1 ) (,) C
)  <->  ( v  e.  RR  /\  ( A  -  1 )  < 
v  /\  v  <  C ) ) )
59 elicc2 10968 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( v  e.  ( A [,] B )  <-> 
( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) )
6059adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,] B
)  <->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <_  B ) ) )
6158, 60anbi12d 692 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  ( ( A  -  1 ) (,) C )  /\  v  e.  ( A [,] B ) )  <->  ( (
v  e.  RR  /\  ( A  -  1
)  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) ) )
6253, 61syl5bb 249 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B ) )  <-> 
( ( v  e.  RR  /\  ( A  -  1 )  < 
v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) ) )
6346, 52, 623bitr4d 277 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,) C
)  <->  v  e.  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) ) )
6463eqrdv 2434 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  =  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) )
65 ineq1 3528 . . . . . . 7  |-  ( v  =  ( ( A  -  1 ) (,) C )  ->  (
v  i^i  ( A [,] B ) )  =  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B ) ) )
6665eqeq2d 2447 . . . . . 6  |-  ( v  =  ( ( A  -  1 ) (,) C )  ->  (
( A [,) C
)  =  ( v  i^i  ( A [,] B ) )  <->  ( A [,) C )  =  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) ) )
6766rspcev 3045 . . . . 5  |-  ( ( ( ( A  - 
1 ) (,) C
)  e.  ( topGen ` 
ran  (,) )  /\  ( A [,) C )  =  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B ) ) )  ->  E. v  e.  ( topGen `  ran  (,) )
( A [,) C
)  =  ( v  i^i  ( A [,] B ) ) )
681, 64, 67sylancr 645 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  E. v  e.  ( topGen `  ran  (,) )
( A [,) C
)  =  ( v  i^i  ( A [,] B ) ) )
69 retop 18788 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
70 ovex 6099 . . . . 5  |-  ( A [,] B )  e. 
_V
71 elrest 13648 . . . . 5  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A [,] B )  e. 
_V )  ->  (
( A [,) C
)  e.  ( (
topGen `  ran  (,) )t  ( A [,] B ) )  <->  E. v  e.  ( topGen `
 ran  (,) )
( A [,) C
)  =  ( v  i^i  ( A [,] B ) ) ) )
7269, 70, 71mp2an 654 . . . 4  |-  ( ( A [,) C )  e.  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )  <->  E. v  e.  ( topGen `  ran  (,) )
( A [,) C
)  =  ( v  i^i  ( A [,] B ) ) )
7368, 72sylibr 204 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  e.  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
74 iccssre 10985 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
7574adantr 452 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,] B )  C_  RR )
76 eqid 2436 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
77 icoopnst.1 . . . . 5  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( A [,] B )  X.  ( A [,] B ) ) ) )
7876, 77resubmet 18826 . . . 4  |-  ( ( A [,] B ) 
C_  RR  ->  J  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
7975, 78syl 16 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  J  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
8073, 79eleqtrrd 2513 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  e.  J
)
8180ex 424 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  ->  ( A [,) C )  e.  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2699   _Vcvv 2949    i^i cin 3312    C_ wss 3313   class class class wbr 4205    X. cxp 4869   ran crn 4872    |` cres 4873    o. ccom 4875   ` cfv 5447  (class class class)co 6074   RRcr 8982   1c1 8984   RR*cxr 9112    < clt 9113    <_ cle 9114    - cmin 9284   (,)cioo 10909   (,]cioc 10910   [,)cico 10911   [,]cicc 10912   abscabs 12032   ↾t crest 13641   topGenctg 13658   MetOpencmopn 16684   Topctop 16951
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 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061
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 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-map 7013  df-en 7103  df-dom 7104  df-sdom 7105  df-sup 7439  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-n0 10215  df-z 10276  df-uz 10482  df-q 10568  df-rp 10606  df-xneg 10703  df-xadd 10704  df-xmul 10705  df-ioo 10913  df-ioc 10914  df-ico 10915  df-icc 10916  df-seq 11317  df-exp 11376  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-rest 13643  df-topgen 13660  df-psmet 16687  df-xmet 16688  df-met 16689  df-bl 16690  df-mopn 16691  df-top 16956  df-bases 16958  df-topon 16959
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