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Theorem ioombl 19447
Description: An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
ioombl  |-  ( A (,) B )  e. 
dom  vol

Proof of Theorem ioombl
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snunioo 11012 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
213expa 1153 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B
) )
32adantrr 698 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
4 lbico1 10955 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  ( A [,) B
) )
543expa 1153 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
65adantrr 698 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  ( A [,) B ) )
76snssd 3935 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 10950 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3533 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  ( { A }  i^i  ( A (,) B ) ) )
11 simpll 731 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  RR* )
12 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  B  e.  RR* )
13 df-icc 10912 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 10909 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 9133 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 10920 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( A [,] A
)  i^i  ( A (,) B ) )  =  (/) )
1711, 11, 12, 16syl3anc 1184 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  (/) )
1810, 17eqtr3d 2469 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3708 . . . . . . . 8  |-  ( ( { A }  C_  ( A [,) B )  /\  ( { A }  i^i  ( A (,) B ) )  =  (/) )  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
207, 18, 19syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
213, 20mpbid 202 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) )
22 mnfxr 10703 . . . . . . . . . 10  |-  -oo  e.  RR*
2322a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  -oo  e.  RR* )
24 simprr 734 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  -oo  <  A
)
25 simprl 733 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  <  B )
26 xrre2 10747 . . . . . . . . 9  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  (  -oo  <  A  /\  A  <  B ) )  ->  A  e.  RR )
2723, 11, 12, 24, 25, 26syl32anc 1192 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  RR )
28 icombl 19446 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  e.  dom  vol )
2927, 12, 28syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A [,) B )  e.  dom  vol )
3027snssd 3935 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 19379 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol * `  { A } )  =  0 )
3227, 31syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( vol * `
 { A }
)  =  0 )
33 nulmbl 19418 . . . . . . . 8  |-  ( ( { A }  C_  RR  /\  ( vol * `  { A } )  =  0 )  ->  { A }  e.  dom  vol )
3430, 32, 33syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  e.  dom  vol )
35 difmbl 19425 . . . . . . 7  |-  ( ( ( A [,) B
)  e.  dom  vol  /\ 
{ A }  e.  dom  vol )  ->  (
( A [,) B
)  \  { A } )  e.  dom  vol )
3629, 34, 35syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  e. 
dom  vol )
3721, 36eqeltrrd 2510 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A (,) B )  e.  dom  vol )
3837expr 599 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <  A  ->  ( A (,) B )  e.  dom  vol ) )
39 uncom 3483 . . . . . . . . 9  |-  ( ( B [,)  +oo )  u.  (  -oo (,) B
) )  =  ( (  -oo (,) B
)  u.  ( B [,)  +oo ) )
4022a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  e.  RR* )
41 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
42 pnfxr 10702 . . . . . . . . . . 11  |-  +oo  e.  RR*
4342a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  +oo  e.  RR* )
44 simpll 731 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
45 mnfle 10718 . . . . . . . . . . . 12  |-  ( A  e.  RR*  ->  -oo  <_  A )
4645ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  <_  A )
47 simpr 448 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
4840, 44, 41, 46, 47xrlelttrd 10739 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  <  B )
49 pnfge 10716 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_  +oo )
5041, 49syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
51 df-ico 10911 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 9132 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 10736 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_  +oo )  ->  w  <  +oo ) )
54 xrltletr 10736 . . . . . . . . . . 11  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
(  -oo  <  B  /\  B  <_  w )  ->  -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 10921 . . . . . . . . . 10  |-  ( ( (  -oo  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  (  -oo  <  B  /\  B  <_  +oo ) )  -> 
( (  -oo (,) B )  u.  ( B [,)  +oo ) )  =  (  -oo (,)  +oo ) )
5640, 41, 43, 48, 50, 55syl32anc 1192 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( (  -oo (,) B )  u.  ( B [,)  +oo ) )  =  (  -oo (,)  +oo ) )
5739, 56syl5eq 2479 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  (  -oo (,)  +oo ) )
58 ioomax 10974 . . . . . . . 8  |-  (  -oo (,) 
+oo )  =  RR
5957, 58syl6eq 2483 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  RR )
60 ssun1 3502 . . . . . . . . 9  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  (  -oo (,) B ) )
6160, 59syl5sseq 3388 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  RR )
62 incom 3525 . . . . . . . . 9  |-  ( ( B [,)  +oo )  i^i  (  -oo (,) B
) )  =  ( (  -oo (,) B
)  i^i  ( B [,)  +oo ) )
6314, 51, 52ixxdisj 10920 . . . . . . . . . 10  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo (,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
6440, 41, 43, 63syl3anc 1184 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( (  -oo (,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
6562, 64syl5eq 2479 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  (  -oo (,) B ) )  =  (/) )
66 uneqdifeq 3708 . . . . . . . 8  |-  ( ( ( B [,)  +oo )  C_  RR  /\  (
( B [,)  +oo )  i^i  (  -oo (,) B ) )  =  (/) )  ->  ( ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  RR  <->  ( RR  \ 
( B [,)  +oo ) )  =  ( 
-oo (,) B ) ) )
6761, 65, 66syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( ( B [,)  +oo )  u.  (  -oo (,) B
) )  =  RR  <->  ( RR  \  ( B [,)  +oo ) )  =  (  -oo (,) B
) ) )
6859, 67mpbid 202 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,)  +oo ) )  =  ( 
-oo (,) B ) )
69 rembl 19423 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 10726 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  +oo  e.  RR* )  ->  ( B  <_  +oo  <->  ( B  <  +oo  \/  B  =  +oo ) ) )
7141, 42, 70sylancl 644 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <_  +oo 
<->  ( B  <  +oo  \/  B  =  +oo )
) )
7250, 71mpbid 202 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  \/  B  =  +oo ) )
73 xrre2 10747 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  ( A  <  B  /\  B  <  +oo ) )  ->  B  e.  RR )
7473expr 599 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
7542, 74mp3anl3 1275 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
7675orim1d 813 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B  <  +oo  \/  B  =  +oo )  ->  ( B  e.  RR  \/  B  =  +oo ) ) )
7772, 76mpd 15 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  e.  RR  \/  B  = 
+oo ) )
78 icombl1 19445 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
79 oveq1 6079 . . . . . . . . . . 11  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
80 pnfge 10716 . . . . . . . . . . . . 13  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
8142, 80ax-mp 8 . . . . . . . . . . . 12  |-  +oo  <_  +oo
82 ico0 10951 . . . . . . . . . . . . 13  |-  ( ( 
+oo  e.  RR*  /\  +oo  e.  RR* )  ->  (
(  +oo [,)  +oo )  =  (/)  <->  +oo  <_  +oo ) )
8342, 42, 82mp2an 654 . . . . . . . . . . . 12  |-  ( ( 
+oo [,)  +oo )  =  (/) 
<-> 
+oo  <_  +oo )
8481, 83mpbir 201 . . . . . . . . . . 11  |-  (  +oo [,) 
+oo )  =  (/)
8579, 84syl6eq 2483 . . . . . . . . . 10  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
86 0mbl 19422 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2523 . . . . . . . . 9  |-  ( B  =  +oo  ->  ( B [,)  +oo )  e.  dom  vol )
8878, 87jaoi 369 . . . . . . . 8  |-  ( ( B  e.  RR  \/  B  =  +oo )  -> 
( B [,)  +oo )  e.  dom  vol )
8977, 88syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  e.  dom  vol )
90 difmbl 19425 . . . . . . 7  |-  ( ( RR  e.  dom  vol  /\  ( B [,)  +oo )  e.  dom  vol )  ->  ( RR  \  ( B [,)  +oo ) )  e. 
dom  vol )
9169, 89, 90sylancr 645 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,)  +oo ) )  e.  dom  vol )
9268, 91eqeltrrd 2510 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo (,) B )  e.  dom  vol )
93 oveq1 6079 . . . . . 6  |-  (  -oo  =  A  ->  (  -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2501 . . . . 5  |-  (  -oo  =  A  ->  ( ( 
-oo (,) B )  e. 
dom  vol  <->  ( A (,) B )  e.  dom  vol ) )
9592, 94syl5ibcom 212 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  =  A  ->  ( A (,) B )  e.  dom  vol ) )
96 xrleloe 10726 . . . . . 6  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo  <_  A  <->  (  -oo  <  A  \/  -oo  =  A ) ) )
9722, 44, 96sylancr 645 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <_  A  <-> 
(  -oo  <  A  \/  -oo  =  A ) ) )
9846, 97mpbid 202 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <  A  \/  -oo  =  A
) )
9938, 95, 98mpjaod 371 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A (,) B )  e.  dom  vol )
100 ioo0 10930 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 9132 . . . . . . 7  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
102101ancoms 440 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
103100, 102bitrd 245 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  -.  A  <  B ) )
104103biimpar 472 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  =  (/) )
105104, 86syl6eqel 2523 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  e.  dom  vol )
10699, 105pm2.61dan 767 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e. 
dom  vol )
107 ndmioo 10932 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2523 . 2  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  e.  dom  vol )
109106, 108pm2.61i 158 1  |-  ( A (,) B )  e. 
dom  vol
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   dom cdm 4869   ` cfv 5445  (class class class)co 6072   RRcr 8978   0cc0 8979    +oocpnf 9106    -oocmnf 9107   RR*cxr 9108    < clt 9109    <_ cle 9110   (,)cioo 10905   [,)cico 10907   [,]cicc 10908   vol *covol 19347   volcvol 19348
This theorem is referenced by:  iccmbl  19448  ovolioo  19450  uniioovol  19459  uniioombllem4  19466  uniioombllem5  19467  opnmblALT  19483  mbfid  19516  ditgcl  19733  ditgswap  19734  ditgsplitlem  19735  ftc1lem1  19907  ftc1lem2  19908  ftc1a  19909  ftc1lem4  19911  ftc2  19916  ftc2ditglem  19917  itgsubstlem  19920  itg2gt0cn  26206  ftc1cnnclem  26224  areacirc  26234  lhe4.4ex1a  27461  ioovolcl  27656  volioo  27657  itgsin0pilem1  27658  iblioosinexp  27661  itgsinexplem1  27662  itgsinexp  27663  wallispilem2  27729
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-q 10564  df-rp 10602  df-xadd 10700  df-ioo 10909  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-clim 12270  df-rlim 12271  df-sum 12468  df-xmet 16683  df-met 16684  df-ovol 19349  df-vol 19350
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