MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ioombl Unicode version

Theorem ioombl 18938
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 10778 . . . . . . . . 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 10722 . . . . . . . . . . 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 3776 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 10717 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3382 . . . . . . . . 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 10679 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 10676 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 8907 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 10687 . . . . . . . . . 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 2330 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3555 . . . . . . . 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 10472 . . . . . . . . . 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 10515 . . . . . . . . 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 18937 . . . . . . . 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 3776 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 18870 . . . . . . . . 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 18909 . . . . . . . 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 18916 . . . . . . 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 2371 . . . . 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 3332 . . . . . . . . 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 10471 . . . . . . . . . . 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 10486 . . . . . . . . . . . 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 10507 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  -oo  <  B )
49 pnfge 10485 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_  +oo )
5041, 49syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
51 df-ico 10678 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 8906 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 10504 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_  +oo )  ->  w  <  +oo ) )
54 xrltletr 10504 . . . . . . . . . . 11  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
(  -oo  <  B  /\  B  <_  w )  ->  -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 10688 . . . . . . . . . 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 2340 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  (  -oo (,)  +oo ) )
58 ioomax 10740 . . . . . . . 8  |-  (  -oo (,) 
+oo )  =  RR
5957, 58syl6eq 2344 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  (  -oo (,) B ) )  =  RR )
60 ssun1 3351 . . . . . . . . 9  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  (  -oo (,) B ) )
6160, 59syl5sseq 3239 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  RR )
62 incom 3374 . . . . . . . . 9  |-  ( ( B [,)  +oo )  i^i  (  -oo (,) B
) )  =  ( (  -oo (,) B
)  i^i  ( B [,)  +oo ) )
6314, 51, 52ixxdisj 10687 . . . . . . . . . 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 2340 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  (  -oo (,) B ) )  =  (/) )
66 uneqdifeq 3555 . . . . . . . 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 18914 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 10494 . . . . . . . . . . 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 10515 . . . . . . . . . . . 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 18936 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
79 oveq1 5881 . . . . . . . . . . 11  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
80 pnfge 10485 . . . . . . . . . . . . 13  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
8142, 80ax-mp 8 . . . . . . . . . . . 12  |-  +oo  <_  +oo
82 ico0 10718 . . . . . . . . . . . . 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 2344 . . . . . . . . . 10  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
86 0mbl 18913 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2384 . . . . . . . . 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 18916 . . . . . . 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 2371 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  (  -oo (,) B )  e.  dom  vol )
93 oveq1 5881 . . . . . 6  |-  (  -oo  =  A  ->  (  -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2362 . . . . 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 10494 . . . . . 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 10697 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 8906 . . . . . . 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 2384 . . 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 10699 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2384 . 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 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   (,)cioo 10672   [,)cico 10674   [,]cicc 10675   vol *covol 18838   volcvol 18839
This theorem is referenced by:  iccmbl  18939  ovolioo  18941  uniioovol  18950  uniioombllem4  18957  uniioombllem5  18958  opnmblALT  18974  mbfid  19007  ditgcl  19224  ditgswap  19225  ditgsplitlem  19226  ftc1lem1  19398  ftc1lem2  19399  ftc1a  19400  ftc1lem4  19402  ftc2  19407  ftc2ditglem  19408  itgsubstlem  19411  itg2gt0cn  25006  ftc1cnnclem  25024  areacirc  25034  lhe4.4ex1a  27649  ioovolcl  27845  volioo  27846  itgsin0pilem1  27847  iblioosinexp  27850  itgsinexplem1  27851  itgsinexp  27852  wallispilem2  27918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841
  Copyright terms: Public domain W3C validator