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Theorem ioombl 19464
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 11028 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
213expa 1154 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B
) )
32adantrr 699 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
4 lbico1 10971 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  ( A [,) B
) )
543expa 1154 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
65adantrr 699 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  ( A [,) B ) )
76snssd 3945 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 10966 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3543 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  ( { A }  i^i  ( A (,) B ) ) )
11 simpll 732 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  RR* )
12 simplr 733 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  B  e.  RR* )
13 df-icc 10928 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 10925 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 9149 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 10936 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( A [,] A
)  i^i  ( A (,) B ) )  =  (/) )
1711, 11, 12, 16syl3anc 1185 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  (/) )
1810, 17eqtr3d 2472 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3718 . . . . . . . 8  |-  ( ( { A }  C_  ( A [,) B )  /\  ( { A }  i^i  ( A (,) B ) )  =  (/) )  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
207, 18, 19syl2anc 644 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
213, 20mpbid 203 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) )
22 mnfxr 10719 . . . . . . . . . 10  |-  -oo  e.  RR*
2322a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  -oo  e.  RR* )
24 simprr 735 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  -oo  <  A
)
25 simprl 734 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  <  B )
26 xrre2 10763 . . . . . . . . 9  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  (  -oo  <  A  /\  A  <  B ) )  ->  A  e.  RR )
2723, 11, 12, 24, 25, 26syl32anc 1193 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  RR )
28 icombl 19463 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  e.  dom  vol )
2927, 12, 28syl2anc 644 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A [,) B )  e.  dom  vol )
3027snssd 3945 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 19396 . . . . . . . . 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 19435 . . . . . . . 8  |-  ( ( { A }  C_  RR  /\  ( vol * `  { A } )  =  0 )  ->  { A }  e.  dom  vol )
3430, 32, 33syl2anc 644 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  e.  dom  vol )
35 difmbl 19442 . . . . . . 7  |-  ( ( ( A [,) B
)  e.  dom  vol  /\ 
{ A }  e.  dom  vol )  ->  (
( A [,) B
)  \  { A } )  e.  dom  vol )
3629, 34, 35syl2anc 644 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  e. 
dom  vol )
3721, 36eqeltrrd 2513 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A (,) B )  e.  dom  vol )
3837expr 600 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <  A  ->  ( A (,) B )  e.  dom  vol ) )
39 uncom 3493 . . . . . . . . 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 733 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
42 pnfxr 10718 . . . . . . . . . . 11  |-  +oo  e.  RR*
4342a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  +oo  e.  RR* )
44 simpll 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
45 mnfle 10734 . . . . . . . . . . . 12  |-  ( A  e.  RR*  ->  -oo  <_  A )
4645ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  <_  A )
47 simpr 449 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
4840, 44, 41, 46, 47xrlelttrd 10755 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  <  B )
49 pnfge 10732 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_  +oo )
5041, 49syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
51 df-ico 10927 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 9148 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 10752 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_  +oo )  ->  w  <  +oo ) )
54 xrltletr 10752 . . . . . . . . . . 11  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
(  -oo  <  B  /\  B  <_  w )  ->  -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 10937 . . . . . . . . . 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 1193 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( (  -oo (,) B )  u.  ( B [,)  +oo ) )  =  (  -oo (,)  +oo ) )
5739, 56syl5eq 2482 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  (  -oo (,)  +oo ) )
58 ioomax 10990 . . . . . . . 8  |-  (  -oo (,) 
+oo )  =  RR
5957, 58syl6eq 2486 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  RR )
60 ssun1 3512 . . . . . . . . 9  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  (  -oo (,) B ) )
6160, 59syl5sseq 3398 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  RR )
62 incom 3535 . . . . . . . . 9  |-  ( ( B [,)  +oo )  i^i  (  -oo (,) B
) )  =  ( (  -oo (,) B
)  i^i  ( B [,)  +oo ) )
6314, 51, 52ixxdisj 10936 . . . . . . . . . 10  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo (,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
6440, 41, 43, 63syl3anc 1185 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( (  -oo (,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
6562, 64syl5eq 2482 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  (  -oo (,) B ) )  =  (/) )
66 uneqdifeq 3718 . . . . . . . 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 644 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( ( B [,)  +oo )  u.  (  -oo (,) B
) )  =  RR  <->  ( RR  \  ( B [,)  +oo ) )  =  (  -oo (,) B
) ) )
6859, 67mpbid 203 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,)  +oo ) )  =  ( 
-oo (,) B ) )
69 rembl 19440 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 10742 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  +oo  e.  RR* )  ->  ( B  <_  +oo  <->  ( B  <  +oo  \/  B  =  +oo ) ) )
7141, 42, 70sylancl 645 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <_  +oo 
<->  ( B  <  +oo  \/  B  =  +oo )
) )
7250, 71mpbid 203 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  \/  B  =  +oo ) )
73 xrre2 10763 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  ( A  <  B  /\  B  <  +oo ) )  ->  B  e.  RR )
7473expr 600 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
7542, 74mp3anl3 1276 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
7675orim1d 814 . . . . . . . . 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 19462 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
79 oveq1 6091 . . . . . . . . . . 11  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
80 pnfge 10732 . . . . . . . . . . . . 13  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
8142, 80ax-mp 5 . . . . . . . . . . . 12  |-  +oo  <_  +oo
82 ico0 10967 . . . . . . . . . . . . 13  |-  ( ( 
+oo  e.  RR*  /\  +oo  e.  RR* )  ->  (
(  +oo [,)  +oo )  =  (/)  <->  +oo  <_  +oo ) )
8342, 42, 82mp2an 655 . . . . . . . . . . . 12  |-  ( ( 
+oo [,)  +oo )  =  (/) 
<-> 
+oo  <_  +oo )
8481, 83mpbir 202 . . . . . . . . . . 11  |-  (  +oo [,) 
+oo )  =  (/)
8579, 84syl6eq 2486 . . . . . . . . . 10  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
86 0mbl 19439 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2526 . . . . . . . . 9  |-  ( B  =  +oo  ->  ( B [,)  +oo )  e.  dom  vol )
8878, 87jaoi 370 . . . . . . . 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 19442 . . . . . . 7  |-  ( ( RR  e.  dom  vol  /\  ( B [,)  +oo )  e.  dom  vol )  ->  ( RR  \  ( B [,)  +oo ) )  e. 
dom  vol )
9169, 89, 90sylancr 646 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,)  +oo ) )  e.  dom  vol )
9268, 91eqeltrrd 2513 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo (,) B )  e.  dom  vol )
93 oveq1 6091 . . . . . 6  |-  (  -oo  =  A  ->  (  -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2504 . . . . 5  |-  (  -oo  =  A  ->  ( ( 
-oo (,) B )  e. 
dom  vol  <->  ( A (,) B )  e.  dom  vol ) )
9592, 94syl5ibcom 213 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  =  A  ->  ( A (,) B )  e.  dom  vol ) )
96 xrleloe 10742 . . . . . 6  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo  <_  A  <->  (  -oo  <  A  \/  -oo  =  A ) ) )
9722, 44, 96sylancr 646 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <_  A  <-> 
(  -oo  <  A  \/  -oo  =  A ) ) )
9846, 97mpbid 203 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <  A  \/  -oo  =  A
) )
9938, 95, 98mpjaod 372 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A (,) B )  e.  dom  vol )
100 ioo0 10946 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 9148 . . . . . . 7  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
102101ancoms 441 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
103100, 102bitrd 246 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  -.  A  <  B ) )
104103biimpar 473 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  =  (/) )
105104, 86syl6eqel 2526 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  e.  dom  vol )
10699, 105pm2.61dan 768 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e. 
dom  vol )
107 ndmioo 10948 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2526 . 2  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  e.  dom  vol )
109106, 108pm2.61i 159 1  |-  ( A (,) B )  e. 
dom  vol
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   class class class wbr 4215   dom cdm 4881   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995    +oocpnf 9122    -oocmnf 9123   RR*cxr 9124    < clt 9125    <_ cle 9126   (,)cioo 10921   [,)cico 10923   [,]cicc 10924   vol *covol 19364   volcvol 19365
This theorem is referenced by:  iccmbl  19465  ovolioo  19467  uniioovol  19476  uniioombllem4  19483  uniioombllem5  19484  opnmblALT  19500  mbfid  19531  ditgcl  19750  ditgswap  19751  ditgsplitlem  19752  ftc1lem1  19924  ftc1lem2  19925  ftc1a  19926  ftc1lem4  19928  ftc2  19933  ftc2ditglem  19934  itgsubstlem  19937  itg2gt0cn  26274  ftc1cnnclem  26292  ftc1anclem7  26300  ftc1anclem8  26301  ftc1anc  26302  ftc2nc  26303  areacirc  26311  lhe4.4ex1a  27537  ioovolcl  27732  volioo  27733  itgsin0pilem1  27734  iblioosinexp  27737  itgsinexplem1  27738  itgsinexp  27739  wallispilem2  27805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xadd 10716  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-xmet 16700  df-met 16701  df-ovol 19366  df-vol 19367
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