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Theorem ioombl 18922
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 10762 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
213expa 1151 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B
) )
32adantrr 697 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
4 lbico1 10706 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  ( A [,) B
) )
543expa 1151 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
65adantrr 697 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  ( A [,) B ) )
76snssd 3760 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 10701 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3369 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  ( { A }  i^i  ( A (,) B ) ) )
11 simpll 730 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  RR* )
12 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  B  e.  RR* )
13 df-icc 10663 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 10660 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 8891 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 10671 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( A [,] A
)  i^i  ( A (,) B ) )  =  (/) )
1711, 11, 12, 16syl3anc 1182 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  (/) )
1810, 17eqtr3d 2317 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3542 . . . . . . . 8  |-  ( ( { A }  C_  ( A [,) B )  /\  ( { A }  i^i  ( A (,) B ) )  =  (/) )  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
207, 18, 19syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
213, 20mpbid 201 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) )
22 mnfxr 10456 . . . . . . . . . 10  |-  -oo  e.  RR*
2322a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  -oo  e.  RR* )
24 simprr 733 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  -oo  <  A
)
25 simprl 732 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  <  B )
26 xrre2 10499 . . . . . . . . 9  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  (  -oo  <  A  /\  A  <  B ) )  ->  A  e.  RR )
2723, 11, 12, 24, 25, 26syl32anc 1190 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  A  e.  RR )
28 icombl 18921 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  e.  dom  vol )
2927, 12, 28syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A [,) B )  e.  dom  vol )
3027snssd 3760 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 18854 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol * `  { A } )  =  0 )
3227, 31syl 15 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( vol * `
 { A }
)  =  0 )
33 nulmbl 18893 . . . . . . . 8  |-  ( ( { A }  C_  RR  /\  ( vol * `  { A } )  =  0 )  ->  { A }  e.  dom  vol )
3430, 32, 33syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  e.  dom  vol )
35 difmbl 18900 . . . . . . 7  |-  ( ( ( A [,) B
)  e.  dom  vol  /\ 
{ A }  e.  dom  vol )  ->  (
( A [,) B
)  \  { A } )  e.  dom  vol )
3629, 34, 35syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  e. 
dom  vol )
3721, 36eqeltrrd 2358 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A (,) B )  e.  dom  vol )
3837expr 598 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <  A  ->  ( A (,) B )  e.  dom  vol ) )
39 uncom 3319 . . . . . . . . 9  |-  ( ( B [,)  +oo )  u.  (  -oo (,) B
) )  =  ( (  -oo (,) B
)  u.  ( B [,)  +oo ) )
4022a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  e.  RR* )
41 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
42 pnfxr 10455 . . . . . . . . . . 11  |-  +oo  e.  RR*
4342a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  +oo  e.  RR* )
44 simpll 730 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
45 mnfle 10470 . . . . . . . . . . . 12  |-  ( A  e.  RR*  ->  -oo  <_  A )
4645ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  <_  A )
47 simpr 447 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
4840, 44, 41, 46, 47xrlelttrd 10491 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  <  B )
49 pnfge 10469 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_  +oo )
5041, 49syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
51 df-ico 10662 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 8890 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 10488 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_  +oo )  ->  w  <  +oo ) )
54 xrltletr 10488 . . . . . . . . . . 11  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
(  -oo  <  B  /\  B  <_  w )  ->  -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 10672 . . . . . . . . . 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 1190 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( (  -oo (,) B )  u.  ( B [,)  +oo ) )  =  (  -oo (,)  +oo ) )
5739, 56syl5eq 2327 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  (  -oo (,)  +oo ) )
58 ioomax 10724 . . . . . . . 8  |-  (  -oo (,) 
+oo )  =  RR
5957, 58syl6eq 2331 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  RR )
60 ssun1 3338 . . . . . . . . 9  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  (  -oo (,) B ) )
6160, 59syl5sseq 3226 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  RR )
62 incom 3361 . . . . . . . . 9  |-  ( ( B [,)  +oo )  i^i  (  -oo (,) B
) )  =  ( (  -oo (,) B
)  i^i  ( B [,)  +oo ) )
6314, 51, 52ixxdisj 10671 . . . . . . . . . 10  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo (,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
6440, 41, 43, 63syl3anc 1182 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( (  -oo (,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
6562, 64syl5eq 2327 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  (  -oo (,) B ) )  =  (/) )
66 uneqdifeq 3542 . . . . . . . 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 642 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( ( B [,)  +oo )  u.  (  -oo (,) B
) )  =  RR  <->  ( RR  \  ( B [,)  +oo ) )  =  (  -oo (,) B
) ) )
6859, 67mpbid 201 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,)  +oo ) )  =  ( 
-oo (,) B ) )
69 rembl 18898 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 10478 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  +oo  e.  RR* )  ->  ( B  <_  +oo  <->  ( B  <  +oo  \/  B  =  +oo ) ) )
7141, 42, 70sylancl 643 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <_  +oo 
<->  ( B  <  +oo  \/  B  =  +oo )
) )
7250, 71mpbid 201 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  \/  B  =  +oo ) )
73 xrre2 10499 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  ( A  <  B  /\  B  <  +oo ) )  ->  B  e.  RR )
7473expr 598 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
7542, 74mp3anl3 1273 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
7675orim1d 812 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B  <  +oo  \/  B  =  +oo )  ->  ( B  e.  RR  \/  B  =  +oo ) ) )
7772, 76mpd 14 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  e.  RR  \/  B  = 
+oo ) )
78 icombl1 18920 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
79 oveq1 5865 . . . . . . . . . . 11  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
80 pnfge 10469 . . . . . . . . . . . . 13  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
8142, 80ax-mp 8 . . . . . . . . . . . 12  |-  +oo  <_  +oo
82 ico0 10702 . . . . . . . . . . . . 13  |-  ( ( 
+oo  e.  RR*  /\  +oo  e.  RR* )  ->  (
(  +oo [,)  +oo )  =  (/)  <->  +oo  <_  +oo ) )
8342, 42, 82mp2an 653 . . . . . . . . . . . 12  |-  ( ( 
+oo [,)  +oo )  =  (/) 
<-> 
+oo  <_  +oo )
8481, 83mpbir 200 . . . . . . . . . . 11  |-  (  +oo [,) 
+oo )  =  (/)
8579, 84syl6eq 2331 . . . . . . . . . 10  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
86 0mbl 18897 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2371 . . . . . . . . 9  |-  ( B  =  +oo  ->  ( B [,)  +oo )  e.  dom  vol )
8878, 87jaoi 368 . . . . . . . 8  |-  ( ( B  e.  RR  \/  B  =  +oo )  -> 
( B [,)  +oo )  e.  dom  vol )
8977, 88syl 15 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  e.  dom  vol )
90 difmbl 18900 . . . . . . 7  |-  ( ( RR  e.  dom  vol  /\  ( B [,)  +oo )  e.  dom  vol )  ->  ( RR  \  ( B [,)  +oo ) )  e. 
dom  vol )
9169, 89, 90sylancr 644 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,)  +oo ) )  e.  dom  vol )
9268, 91eqeltrrd 2358 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo (,) B )  e.  dom  vol )
93 oveq1 5865 . . . . . 6  |-  (  -oo  =  A  ->  (  -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2349 . . . . 5  |-  (  -oo  =  A  ->  ( ( 
-oo (,) B )  e. 
dom  vol  <->  ( A (,) B )  e.  dom  vol ) )
9592, 94syl5ibcom 211 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  =  A  ->  ( A (,) B )  e.  dom  vol ) )
96 xrleloe 10478 . . . . . 6  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo  <_  A  <->  (  -oo  <  A  \/  -oo  =  A ) ) )
9722, 44, 96sylancr 644 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <_  A  <-> 
(  -oo  <  A  \/  -oo  =  A ) ) )
9846, 97mpbid 201 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo  <  A  \/  -oo  =  A
) )
9938, 95, 98mpjaod 370 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A (,) B )  e.  dom  vol )
100 ioo0 10681 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 8890 . . . . . . 7  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
102101ancoms 439 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
103100, 102bitrd 244 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  -.  A  <  B ) )
104103biimpar 471 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  =  (/) )
105104, 86syl6eqel 2371 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  e.  dom  vol )
10699, 105pm2.61dan 766 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e. 
dom  vol )
107 ndmioo 10683 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2371 . 2  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  e.  dom  vol )
109106, 108pm2.61i 156 1  |-  ( A (,) B )  e. 
dom  vol
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656   [,)cico 10658   [,]cicc 10659   vol *covol 18822   volcvol 18823
This theorem is referenced by:  iccmbl  18923  ovolioo  18925  uniioovol  18934  uniioombllem4  18941  uniioombllem5  18942  opnmblALT  18958  mbfid  18991  ditgcl  19208  ditgswap  19209  ditgsplitlem  19210  ftc1lem1  19382  ftc1lem2  19383  ftc1a  19384  ftc1lem4  19386  ftc2  19391  ftc2ditglem  19392  itgsubstlem  19395  areacirc  24931  lhe4.4ex1a  27546  ioovolcl  27742  volioo  27743  itgsin0pilem1  27744  iblioosinexp  27747  itgsinexplem1  27748  itgsinexp  27749  wallispilem2  27815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825
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