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Theorem icoopnst 20486
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 20320 . . . . 5  |-  ( ( A  -  1 ) (,) C )  e.  ( topGen `  ran  (,) )
2 simp1 988 . . . . . . . . . . 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 10161 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  ( A  -  1 )  <  A )
54adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  -  1 )  <  A )
6 peano2rem 9667 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
76adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  -  1 )  e.  RR )
8 ltletr 9458 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  -  1 )  e.  RR  /\  A  e.  RR  /\  v  e.  RR )  ->  (
( ( A  - 
1 )  <  A  /\  A  <_  v )  ->  ( A  - 
1 )  <  v
) )
983expb 1188 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  -  1 )  e.  RR  /\  ( A  e.  RR  /\  v  e.  RR ) )  ->  ( (
( A  -  1 )  <  A  /\  A  <_  v )  -> 
( A  -  1 )  <  v ) )
107, 9mpancom 669 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( ( ( A  -  1 )  < 
A  /\  A  <_  v )  ->  ( A  -  1 )  < 
v ) )
115, 10mpand 675 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  <_  v  ->  ( A  -  1 )  <  v ) )
1211impr 619 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( v  e.  RR  /\  A  <_  v )
)  ->  ( A  -  1 )  < 
v )
13123adantr3 1149 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <  C ) )  ->  ( A  - 
1 )  <  v
)
1413ex 434 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  (
( v  e.  RR  /\  A  <_  v  /\  v  <  C )  -> 
( A  -  1 )  <  v ) )
1514ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  ( A  -  1 )  <  v ) )
16 simp3 990 . . . . . . . . . . 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 1169 . . . . . . . . 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 989 . . . . . . . . . . 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 9421 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  A  e.  RR* )
22 elioc2 11350 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
2321, 22sylan 471 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <-> 
( C  e.  RR  /\  A  <  C  /\  C  <_  B ) ) )
2423biimpa 484 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) )
25 ltletr 9458 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
( v  <  C  /\  C  <_  B )  ->  v  <  B
) )
26 ltle 9455 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( v  e.  RR  /\  B  e.  RR )  ->  ( v  <  B  ->  v  <_  B )
)
27263adant2 1007 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
v  <  B  ->  v  <_  B ) )
2825, 27syld 44 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
( v  <  C  /\  C  <_  B )  ->  v  <_  B
) )
29283expa 1187 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( v  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( v  <  C  /\  C  <_  B )  ->  v  <_  B ) )
3029an31s 804 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  ->  ( ( v  <  C  /\  C  <_  B )  ->  v  <_  B ) )
3130imp 429 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  /\  (
v  <  C  /\  C  <_  B ) )  ->  v  <_  B
)
3231ancom2s 800 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  /\  ( C  <_  B  /\  v  <  C ) )  -> 
v  <_  B )
3332an4s 822 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B )  /\  (
v  e.  RR  /\  v  <  C ) )  ->  v  <_  B
)
34333adantr2 1148 . . . . . . . . . . . . . . 15  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B )  /\  (
v  e.  RR  /\  A  <_  v  /\  v  <  C ) )  -> 
v  <_  B )
3534ex 434 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B
)  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
3635anasss 647 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  C  <_  B )
)  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
37363adantr2 1148 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  A  <  C  /\  C  <_  B ) )  ->  ( ( v  e.  RR  /\  A  <_  v  /\  v  < 
C )  ->  v  <_  B ) )
3837adantll 713 . . . . . . . . . . 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 470 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
403, 20, 393jcad 1169 . . . . . . . . 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 533 . . . . . . . 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 991 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  v  e.  RR )
43 simpr2 995 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  A  <_  v
)
44 simpl3 993 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  v  <  C
)
4542, 43, 443jca 1168 . . . . . . . 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 203 . . . . . . 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 753 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  A  e.  RR )
4824simp1d 1000 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  C  e.  RR )
49 rexr 9421 . . . . . . . . 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 11351 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR* )  -> 
( v  e.  ( A [,) C )  <-> 
( v  e.  RR  /\  A  <_  v  /\  v  <  C ) ) )
5247, 50, 51syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,) C
)  <->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <  C ) ) )
53 elin 3534 . . . . . . . 8  |-  ( v  e.  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B
) )  <->  ( v  e.  ( ( A  - 
1 ) (,) C
)  /\  v  e.  ( A [,] B ) ) )
54 rexr 9421 . . . . . . . . . . . 12  |-  ( ( A  -  1 )  e.  RR  ->  ( A  -  1 )  e.  RR* )
556, 54syl 16 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR* )
5655ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A  -  1 )  e. 
RR* )
57 elioo2 11333 . . . . . . . . . 10  |-  ( ( ( A  -  1 )  e.  RR*  /\  C  e.  RR* )  ->  (
v  e.  ( ( A  -  1 ) (,) C )  <->  ( v  e.  RR  /\  ( A  -  1 )  < 
v  /\  v  <  C ) ) )
5856, 50, 57syl2anc 661 . . . . . . . . 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 11352 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( v  e.  ( A [,] B )  <-> 
( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) )
6059adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,] B
)  <->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <_  B ) ) )
6158, 60anbi12d 710 . . . . . . . 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 257 . . . . . . 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 285 . . . . . 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 2436 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  =  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) )
65 ineq1 3540 . . . . . . 7  |-  ( v  =  ( ( A  -  1 ) (,) C )  ->  (
v  i^i  ( A [,] B ) )  =  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B ) ) )
6665eqeq2d 2449 . . . . . 6  |-  ( v  =  ( ( A  -  1 ) (,) C )  ->  (
( A [,) C
)  =  ( v  i^i  ( A [,] B ) )  <->  ( A [,) C )  =  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) ) )
6766rspcev 3068 . . . . 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 663 . . . 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 20315 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
70 ovex 6111 . . . . 5  |-  ( A [,] B )  e. 
_V
71 elrest 14358 . . . . 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 672 . . . 4  |-  ( ( A [,) C )  e.  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )  <->  E. v  e.  ( topGen `  ran  (,) )
( A [,) C
)  =  ( v  i^i  ( A [,] B ) ) )
7368, 72sylibr 212 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  e.  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
74 iccssre 11369 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
7574adantr 465 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,] B )  C_  RR )
76 eqid 2438 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
77 icoopnst.1 . . . . 5  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( A [,] B )  X.  ( A [,] B ) ) ) )
7876, 77resubmet 20354 . . . 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 2515 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  e.  J
)
8180ex 434 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  ->  ( A [,) C )  e.  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2711   _Vcvv 2967    i^i cin 3322    C_ wss 3323   class class class wbr 4287    X. cxp 4833   ran crn 4836    |` cres 4837    o. ccom 4839   ` cfv 5413  (class class class)co 6086   RRcr 9273   1c1 9275   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587   (,)cioo 11292   (,]cioc 11293   [,)cico 11294   [,]cicc 11295   abscabs 12715   ↾t crest 14351   topGenctg 14368   MetOpencmopn 17781   Topctop 18473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-rest 14353  df-topgen 14374  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-top 18478  df-bases 18480  df-topon 18481
This theorem is referenced by: (None)
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