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Theorem ioombl 19327
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 10956 . . . . . . . . 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 10899 . . . . . . . . . . 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 3887 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 10894 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3485 . . . . . . . . 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 10856 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 10853 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 9078 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 10864 . . . . . . . . . 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 2422 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3660 . . . . . . . 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 10647 . . . . . . . . . 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 10691 . . . . . . . . 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 19326 . . . . . . . 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 3887 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 19259 . . . . . . . . 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 19298 . . . . . . . 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 19305 . . . . . . 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 2463 . . . . 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 3435 . . . . . . . . 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 10646 . . . . . . . . . . 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 10662 . . . . . . . . . . . 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 10683 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  <  B )
49 pnfge 10660 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_  +oo )
5041, 49syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
51 df-ico 10855 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 9077 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 10680 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_  +oo )  ->  w  <  +oo ) )
54 xrltletr 10680 . . . . . . . . . . 11  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
(  -oo  <  B  /\  B  <_  w )  ->  -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 10865 . . . . . . . . . 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 2432 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  (  -oo (,)  +oo ) )
58 ioomax 10918 . . . . . . . 8  |-  (  -oo (,) 
+oo )  =  RR
5957, 58syl6eq 2436 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  RR )
60 ssun1 3454 . . . . . . . . 9  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  (  -oo (,) B ) )
6160, 59syl5sseq 3340 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  RR )
62 incom 3477 . . . . . . . . 9  |-  ( ( B [,)  +oo )  i^i  (  -oo (,) B
) )  =  ( (  -oo (,) B
)  i^i  ( B [,)  +oo ) )
6314, 51, 52ixxdisj 10864 . . . . . . . . . 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 2432 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  (  -oo (,) B ) )  =  (/) )
66 uneqdifeq 3660 . . . . . . . 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 19303 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 10670 . . . . . . . . . . 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 10691 . . . . . . . . . . . 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 19325 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
79 oveq1 6028 . . . . . . . . . . 11  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
80 pnfge 10660 . . . . . . . . . . . . 13  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
8142, 80ax-mp 8 . . . . . . . . . . . 12  |-  +oo  <_  +oo
82 ico0 10895 . . . . . . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
86 0mbl 19302 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2476 . . . . . . . . 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 19305 . . . . . . 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 2463 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo (,) B )  e.  dom  vol )
93 oveq1 6028 . . . . . 6  |-  (  -oo  =  A  ->  (  -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2454 . . . . 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 10670 . . . . . 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 10874 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 9077 . . . . . . 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 2476 . . 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 10876 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2476 . 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 1649    e. wcel 1717    \ cdif 3261    u. cun 3262    i^i cin 3263    C_ wss 3264   (/)c0 3572   {csn 3758   class class class wbr 4154   dom cdm 4819   ` cfv 5395  (class class class)co 6021   RRcr 8923   0cc0 8924    +oocpnf 9051    -oocmnf 9052   RR*cxr 9053    < clt 9054    <_ cle 9055   (,)cioo 10849   [,)cico 10851   [,]cicc 10852   vol *covol 19227   volcvol 19228
This theorem is referenced by:  iccmbl  19328  ovolioo  19330  uniioovol  19339  uniioombllem4  19346  uniioombllem5  19347  opnmblALT  19363  mbfid  19396  ditgcl  19613  ditgswap  19614  ditgsplitlem  19615  ftc1lem1  19787  ftc1lem2  19788  ftc1a  19789  ftc1lem4  19791  ftc2  19796  ftc2ditglem  19797  itgsubstlem  19800  itg2gt0cn  25961  ftc1cnnclem  25979  areacirc  25989  lhe4.4ex1a  27216  ioovolcl  27411  volioo  27412  itgsin0pilem1  27413  iblioosinexp  27416  itgsinexplem1  27417  itgsinexp  27418  wallispilem2  27484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-xadd 10644  df-ioo 10853  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-xmet 16620  df-met 16621  df-ovol 19229  df-vol 19230
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