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Theorem icoopnst 18437
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 18275 . . . . 5  |-  ( ( A  -  1 ) (,) C )  e.  ( topGen `  ran  (,) )
2 simp1 955 . . . . . . . . . . 11  |-  ( ( v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  e.  RR )
32a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  e.  RR ) )
4 ltm1 9596 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  ( A  -  1 )  <  A )
54adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  -  1 )  <  A )
6 peano2rem 9113 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
76adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  -  1 )  e.  RR )
8 ltletr 8913 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  -  1 )  e.  RR  /\  A  e.  RR  /\  v  e.  RR )  ->  (
( ( A  - 
1 )  <  A  /\  A  <_  v )  ->  ( A  - 
1 )  <  v
) )
983expb 1152 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  -  1 )  e.  RR  /\  ( A  e.  RR  /\  v  e.  RR ) )  ->  ( (
( A  -  1 )  <  A  /\  A  <_  v )  -> 
( A  -  1 )  <  v ) )
107, 9mpancom 650 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( ( ( A  -  1 )  < 
A  /\  A  <_  v )  ->  ( A  -  1 )  < 
v ) )
115, 10mpand 656 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  <_  v  ->  ( A  -  1 )  <  v ) )
1211impr 602 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( v  e.  RR  /\  A  <_  v )
)  ->  ( A  -  1 )  < 
v )
13123adantr3 1116 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <  C ) )  ->  ( A  - 
1 )  <  v
)
1413ex 423 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  (
( v  e.  RR  /\  A  <_  v  /\  v  <  C )  -> 
( A  -  1 )  <  v ) )
1514ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  ( A  -  1 )  <  v ) )
16 simp3 957 . . . . . . . . . . 11  |-  ( ( v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <  C )
1716a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <  C ) )
183, 15, 173jcad 1133 . . . . . . . . 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 956 . . . . . . . . . . 11  |-  ( ( v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  A  <_  v )
2019a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  A  <_  v ) )
21 rexr 8877 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  A  e.  RR* )
22 elioc2 10713 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
2321, 22sylan 457 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <-> 
( C  e.  RR  /\  A  <  C  /\  C  <_  B ) ) )
2423biimpa 470 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) )
25 ltletr 8913 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
( v  <  C  /\  C  <_  B )  ->  v  <  B
) )
26 ltle 8910 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( v  e.  RR  /\  B  e.  RR )  ->  ( v  <  B  ->  v  <_  B )
)
27263adant2 974 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
v  <  B  ->  v  <_  B ) )
2825, 27syld 40 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
( v  <  C  /\  C  <_  B )  ->  v  <_  B
) )
29283expa 1151 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( v  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( v  <  C  /\  C  <_  B )  ->  v  <_  B ) )
3029an31s 781 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  ->  ( ( v  <  C  /\  C  <_  B )  ->  v  <_  B ) )
3130imp 418 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  /\  (
v  <  C  /\  C  <_  B ) )  ->  v  <_  B
)
3231ancom2s 777 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  /\  ( C  <_  B  /\  v  <  C ) )  -> 
v  <_  B )
3332an4s 799 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B )  /\  (
v  e.  RR  /\  v  <  C ) )  ->  v  <_  B
)
34333adantr2 1115 . . . . . . . . . . . . . . 15  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B )  /\  (
v  e.  RR  /\  A  <_  v  /\  v  <  C ) )  -> 
v  <_  B )
3534ex 423 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B
)  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
3635anasss 628 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  C  <_  B )
)  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
37363adantr2 1115 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  A  <  C  /\  C  <_  B ) )  ->  ( ( v  e.  RR  /\  A  <_  v  /\  v  < 
C )  ->  v  <_  B ) )
3837adantll 694 . . . . . . . . . . 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 456 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
403, 20, 393jcad 1133 . . . . . . . . 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 519 . . . . . . . 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 958 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  v  e.  RR )
43 simpr2 962 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  A  <_  v
)
44 simpl3 960 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  v  <  C
)
4542, 43, 443jca 1132 . . . . . . . 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 194 . . . . . . 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 730 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  A  e.  RR )
4824simp1d 967 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  C  e.  RR )
49 rexr 8877 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  RR* )
5048, 49syl 15 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  C  e.  RR* )
51 elico2 10714 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR* )  -> 
( v  e.  ( A [,) C )  <-> 
( v  e.  RR  /\  A  <_  v  /\  v  <  C ) ) )
5247, 50, 51syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,) C
)  <->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <  C ) ) )
53 elin 3358 . . . . . . . 8  |-  ( v  e.  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B
) )  <->  ( v  e.  ( ( A  - 
1 ) (,) C
)  /\  v  e.  ( A [,] B ) ) )
54 rexr 8877 . . . . . . . . . . . 12  |-  ( ( A  -  1 )  e.  RR  ->  ( A  -  1 )  e.  RR* )
556, 54syl 15 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR* )
5655ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A  -  1 )  e. 
RR* )
57 elioo2 10697 . . . . . . . . . 10  |-  ( ( ( A  -  1 )  e.  RR*  /\  C  e.  RR* )  ->  (
v  e.  ( ( A  -  1 ) (,) C )  <->  ( v  e.  RR  /\  ( A  -  1 )  < 
v  /\  v  <  C ) ) )
5856, 50, 57syl2anc 642 . . . . . . . . 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 10715 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( v  e.  ( A [,] B )  <-> 
( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) )
6059adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,] B
)  <->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <_  B ) ) )
6158, 60anbi12d 691 . . . . . . . 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 248 . . . . . . 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 276 . . . . . 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 2281 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  =  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) )
65 ineq1 3363 . . . . . . 7  |-  ( v  =  ( ( A  -  1 ) (,) C )  ->  (
v  i^i  ( A [,] B ) )  =  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B ) ) )
6665eqeq2d 2294 . . . . . 6  |-  ( v  =  ( ( A  -  1 ) (,) C )  ->  (
( A [,) C
)  =  ( v  i^i  ( A [,] B ) )  <->  ( A [,) C )  =  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) ) )
6766rspcev 2884 . . . . 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 644 . . . 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 18270 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
70 ovex 5883 . . . . 5  |-  ( A [,] B )  e. 
_V
71 elrest 13332 . . . . 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 653 . . . 4  |-  ( ( A [,) C )  e.  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )  <->  E. v  e.  ( topGen `  ran  (,) )
( A [,) C
)  =  ( v  i^i  ( A [,] B ) ) )
7368, 72sylibr 203 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  e.  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
74 iccssre 10731 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
7574adantr 451 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,] B )  C_  RR )
76 eqid 2283 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
77 icoopnst.1 . . . . 5  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( A [,] B )  X.  ( A [,] B ) ) ) )
7876, 77resubmet 18308 . . . 4  |-  ( ( A [,] B ) 
C_  RR  ->  J  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
7975, 78syl 15 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  J  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
8073, 79eleqtrrd 2360 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  e.  J
)
8180ex 423 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   class class class wbr 4023    X. cxp 4687   ran crn 4690    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   (,)cioo 10656   (,]cioc 10657   [,)cico 10658   [,]cicc 10659   abscabs 11719   ↾t crest 13325   topGenctg 13342   MetOpencmopn 16372   Topctop 16631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639
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