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Theorem icoopnst 21565
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 21399 . . . . 5  |-  ( ( A  -  1 ) (,) C )  e.  ( topGen `  ran  (,) )
2 simp1 996 . . . . . . . . . . 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 10403 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  ( A  -  1 )  <  A )
54adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  -  1 )  <  A )
6 peano2rem 9905 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
76adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  v  e.  RR )  ->  ( A  -  1 )  e.  RR )
8 ltletr 9693 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  -  1 )  e.  RR  /\  A  e.  RR  /\  v  e.  RR )  ->  (
( ( A  - 
1 )  <  A  /\  A  <_  v )  ->  ( A  - 
1 )  <  v
) )
983expb 1197 . . . . . . . . . . . . . . . 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 1157 . . . . . . . . . . . 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 998 . . . . . . . . . . 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 1177 . . . . . . . . 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 997 . . . . . . . . . . 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 9656 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  A  e.  RR* )
22 elioc2 11612 . . . . . . . . . . . . 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 ltleletr 9694 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( v  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
( v  <  C  /\  C  <_  B )  ->  v  <_  B
) )
26253expa 1196 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( v  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( v  <  C  /\  C  <_  B )  ->  v  <_  B ) )
2726an31s 806 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  ->  ( ( v  <  C  /\  C  <_  B )  ->  v  <_  B ) )
2827imp 429 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  /\  (
v  <  C  /\  C  <_  B ) )  ->  v  <_  B
)
2928ancom2s 802 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  v  e.  RR )  /\  ( C  <_  B  /\  v  <  C ) )  -> 
v  <_  B )
3029an4s 826 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B )  /\  (
v  e.  RR  /\  v  <  C ) )  ->  v  <_  B
)
31303adantr2 1156 . . . . . . . . . . . . . . 15  |-  ( ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B )  /\  (
v  e.  RR  /\  A  <_  v  /\  v  <  C ) )  -> 
v  <_  B )
3231ex 434 . . . . . . . . . . . . . 14  |-  ( ( ( B  e.  RR  /\  C  e.  RR )  /\  C  <_  B
)  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
3332anasss 647 . . . . . . . . . . . . 13  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  C  <_  B )
)  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
34333adantr2 1156 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  A  <  C  /\  C  <_  B ) )  ->  ( ( v  e.  RR  /\  A  <_  v  /\  v  < 
C )  ->  v  <_  B ) )
3534adantll 713 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) )  ->  (
( v  e.  RR  /\  A  <_  v  /\  v  <  C )  -> 
v  <_  B )
)
3624, 35syldan 470 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( (
v  e.  RR  /\  A  <_  v  /\  v  <  C )  ->  v  <_  B ) )
373, 20, 363jcad 1177 . . . . . . . . 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 ) ) )
3818, 37jcad 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 ) ) ) )
39 simpl1 999 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  v  e.  RR )
40 simpr2 1003 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  A  <_  v
)
41 simpl3 1001 . . . . . . . . 9  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  v  <  C
)
4239, 40, 413jca 1176 . . . . . . . 8  |-  ( ( ( v  e.  RR  /\  ( A  -  1 )  <  v  /\  v  <  C )  /\  ( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) )  ->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <  C ) )
4338, 42impbid1 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 ) ) ) )
44 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  A  e.  RR )
4524simp1d 1008 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  C  e.  RR )
4645rexrd 9660 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  C  e.  RR* )
47 elico2 11613 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR* )  -> 
( v  e.  ( A [,) C )  <-> 
( v  e.  RR  /\  A  <_  v  /\  v  <  C ) ) )
4844, 46, 47syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,) C
)  <->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <  C ) ) )
49 elin 3683 . . . . . . . 8  |-  ( v  e.  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B
) )  <->  ( v  e.  ( ( A  - 
1 ) (,) C
)  /\  v  e.  ( A [,] B ) ) )
506rexrd 9660 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR* )
5150ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A  -  1 )  e. 
RR* )
52 elioo2 11595 . . . . . . . . . 10  |-  ( ( ( A  -  1 )  e.  RR*  /\  C  e.  RR* )  ->  (
v  e.  ( ( A  -  1 ) (,) C )  <->  ( v  e.  RR  /\  ( A  -  1 )  < 
v  /\  v  <  C ) ) )
5351, 46, 52syl2anc 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 ) ) )
54 elicc2 11614 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( v  e.  ( A [,] B )  <-> 
( v  e.  RR  /\  A  <_  v  /\  v  <_  B ) ) )
5554adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( v  e.  ( A [,] B
)  <->  ( v  e.  RR  /\  A  <_ 
v  /\  v  <_  B ) ) )
5653, 55anbi12d 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 ) ) ) )
5749, 56syl5bb 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 ) ) ) )
5843, 48, 573bitr4d 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 ) ) ) )
5958eqrdv 2454 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  =  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) )
60 ineq1 3689 . . . . . . 7  |-  ( v  =  ( ( A  -  1 ) (,) C )  ->  (
v  i^i  ( A [,] B ) )  =  ( ( ( A  -  1 ) (,) C )  i^i  ( A [,] B ) ) )
6160eqeq2d 2471 . . . . . 6  |-  ( v  =  ( ( A  -  1 ) (,) C )  ->  (
( A [,) C
)  =  ( v  i^i  ( A [,] B ) )  <->  ( A [,) C )  =  ( ( ( A  - 
1 ) (,) C
)  i^i  ( A [,] B ) ) ) )
6261rspcev 3210 . . . . 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 ) ) )
631, 59, 62sylancr 663 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  E. v  e.  ( topGen `  ran  (,) )
( A [,) C
)  =  ( v  i^i  ( A [,] B ) ) )
64 retop 21394 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
65 ovex 6324 . . . . 5  |-  ( A [,] B )  e. 
_V
66 elrest 14845 . . . . 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 ) ) ) )
6764, 65, 66mp2an 672 . . . 4  |-  ( ( A [,) C )  e.  ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )  <->  E. v  e.  ( topGen `  ran  (,) )
( A [,) C
)  =  ( v  i^i  ( A [,] B ) ) )
6863, 67sylibr 212 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  e.  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
69 iccssre 11631 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
7069adantr 465 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,] B )  C_  RR )
71 eqid 2457 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
72 icoopnst.1 . . . . 5  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( A [,] B )  X.  ( A [,] B ) ) ) )
7371, 72resubmet 21433 . . . 4  |-  ( ( A [,] B ) 
C_  RR  ->  J  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
7470, 73syl 16 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  J  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
7568, 74eleqtrrd 2548 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A (,] B ) )  ->  ( A [,) C )  e.  J
)
7675ex 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 973    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   class class class wbr 4456    X. cxp 5006   ran crn 5009    |` cres 5010    o. ccom 5012   ` cfv 5594  (class class class)co 6296   RRcr 9508   1c1 9510   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824   (,)cioo 11554   (,]cioc 11555   [,)cico 11556   [,]cicc 11557   abscabs 13079   ↾t crest 14838   topGenctg 14855   MetOpencmopn 18535   Topctop 19521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-rest 14840  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529
This theorem is referenced by: (None)
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