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Theorem ioombl 22265
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 11698 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
213expa 1197 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B
) )
32adantrr 715 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
4 lbico1 11631 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  ( A [,) B
) )
543expa 1197 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
65adantrr 715 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  e.  ( A [,) B ) )
76snssd 4116 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 11626 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 724 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3639 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  ( { A }  i^i  ( A (,) B ) ) )
11 simpll 752 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  e.  RR* )
12 simplr 754 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  B  e.  RR* )
13 df-icc 11588 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 11585 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 9682 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 11596 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( A [,] A
)  i^i  ( A (,) B ) )  =  (/) )
1711, 11, 12, 16syl3anc 1230 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  (/) )
1810, 17eqtr3d 2445 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3859 . . . . . . . 8  |-  ( ( { A }  C_  ( A [,) B )  /\  ( { A }  i^i  ( A (,) B ) )  =  (/) )  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
207, 18, 19syl2anc 659 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
213, 20mpbid 210 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) )
22 mnfxr 11375 . . . . . . . . . 10  |- -oo  e.  RR*
2322a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  -> -oo  e.  RR* )
24 simprr 758 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  -> -oo  <  A
)
25 simprl 756 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  <  B )
26 xrre2 11423 . . . . . . . . 9  |-  ( ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  ( -oo  <  A  /\  A  <  B ) )  ->  A  e.  RR )
2723, 11, 12, 24, 25, 26syl32anc 1238 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  e.  RR )
28 icombl 22264 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  e.  dom  vol )
2927, 12, 28syl2anc 659 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( A [,) B )  e.  dom  vol )
3027snssd 4116 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 22196 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol* `  { A } )  =  0 )
3227, 31syl 17 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( vol* `  { A }
)  =  0 )
33 nulmbl 22236 . . . . . . . 8  |-  ( ( { A }  C_  RR  /\  ( vol* `  { A } )  =  0 )  ->  { A }  e.  dom  vol )
3430, 32, 33syl2anc 659 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  e.  dom  vol )
35 difmbl 22243 . . . . . . 7  |-  ( ( ( A [,) B
)  e.  dom  vol  /\ 
{ A }  e.  dom  vol )  ->  (
( A [,) B
)  \  { A } )  e.  dom  vol )
3629, 34, 35syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  e. 
dom  vol )
3721, 36eqeltrrd 2491 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( A (,) B )  e.  dom  vol )
3837expr 613 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo  <  A  ->  ( A (,) B )  e.  dom  vol ) )
39 uncom 3586 . . . . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
42 pnfxr 11373 . . . . . . . . . . 11  |- +oo  e.  RR*
4342a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  -> +oo  e.  RR* )
44 simpll 752 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
45 mnfle 11394 . . . . . . . . . . . 12  |-  ( A  e.  RR*  -> -oo  <_  A )
4645ad2antrr 724 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  -> -oo  <_  A )
47 simpr 459 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
4840, 44, 41, 46, 47xrlelttrd 11415 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  -> -oo  <  B )
49 pnfge 11391 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_ +oo )
5041, 49syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_ +oo )
51 df-ico 11587 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 9681 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 11412 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_ +oo )  ->  w  < +oo ) )
54 xrltletr 11412 . . . . . . . . . . 11  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  w )  -> -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 11597 . . . . . . . . . 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 1238 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( -oo (,) B )  u.  ( B [,) +oo ) )  =  ( -oo (,) +oo ) )
5739, 56syl5eq 2455 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  ( -oo (,) +oo )
)
58 ioomax 11651 . . . . . . . 8  |-  ( -oo (,) +oo )  =  RR
5957, 58syl6eq 2459 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  RR )
60 ssun1 3605 . . . . . . . . 9  |-  ( B [,) +oo )  C_  ( ( B [,) +oo )  u.  ( -oo (,) B ) )
6160, 59syl5sseq 3489 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,) +oo )  C_  RR )
62 incom 3631 . . . . . . . . 9  |-  ( ( B [,) +oo )  i^i  ( -oo (,) B
) )  =  ( ( -oo (,) B
)  i^i  ( B [,) +oo ) )
6314, 51, 52ixxdisj 11596 . . . . . . . . . 10  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo (,) B )  i^i  ( B [,) +oo ) )  =  (/) )
6440, 41, 43, 63syl3anc 1230 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( -oo (,) B )  i^i  ( B [,) +oo ) )  =  (/) )
6562, 64syl5eq 2455 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  i^i  ( -oo (,) B
) )  =  (/) )
66 uneqdifeq 3859 . . . . . . . 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 659 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  RR  <->  ( RR  \  ( B [,) +oo ) )  =  ( -oo (,) B ) ) )
6859, 67mpbid 210 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,) +oo ) )  =  ( -oo (,) B ) )
69 rembl 22241 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 11402 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\ +oo  e.  RR* )  ->  ( B  <_ +oo  <->  ( B  < +oo  \/  B  = +oo ) ) )
7141, 42, 70sylancl 660 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <_ +oo 
<->  ( B  < +oo  \/  B  = +oo ) ) )
7250, 71mpbid 210 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  \/  B  = +oo ) )
73 xrre2 11423 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  /\  ( A  <  B  /\  B  < +oo ) )  ->  B  e.  RR )
7473expr 613 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
7542, 74mp3anl3 1322 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
7675orim1d 840 . . . . . . . . 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 22263 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,) +oo )  e. 
dom  vol )
79 oveq1 6284 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( B [,) +oo )  =  ( +oo [,) +oo ) )
80 pnfge 11391 . . . . . . . . . . . . 13  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
8142, 80ax-mp 5 . . . . . . . . . . . 12  |- +oo  <_ +oo
82 ico0 11627 . . . . . . . . . . . . 13  |-  ( ( +oo  e.  RR*  /\ +oo  e.  RR* )  ->  (
( +oo [,) +oo )  =  (/)  <-> +oo  <_ +oo )
)
8342, 42, 82mp2an 670 . . . . . . . . . . . 12  |-  ( ( +oo [,) +oo )  =  (/)  <-> +oo  <_ +oo )
8481, 83mpbir 209 . . . . . . . . . . 11  |-  ( +oo [,) +oo )  =  (/)
8579, 84syl6eq 2459 . . . . . . . . . 10  |-  ( B  = +oo  ->  ( B [,) +oo )  =  (/) )
86 0mbl 22240 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2498 . . . . . . . . 9  |-  ( B  = +oo  ->  ( B [,) +oo )  e. 
dom  vol )
8878, 87jaoi 377 . . . . . . . 8  |-  ( ( B  e.  RR  \/  B  = +oo )  ->  ( B [,) +oo )  e.  dom  vol )
8977, 88syl 17 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,) +oo )  e.  dom  vol )
90 difmbl 22243 . . . . . . 7  |-  ( ( RR  e.  dom  vol  /\  ( B [,) +oo )  e.  dom  vol )  ->  ( RR  \  ( B [,) +oo ) )  e.  dom  vol )
9169, 89, 90sylancr 661 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,) +oo ) )  e.  dom  vol )
9268, 91eqeltrrd 2491 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo (,) B )  e.  dom  vol )
93 oveq1 6284 . . . . . 6  |-  ( -oo  =  A  ->  ( -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2471 . . . . 5  |-  ( -oo  =  A  ->  ( ( -oo (,) B )  e.  dom  vol  <->  ( A (,) B )  e.  dom  vol ) )
9592, 94syl5ibcom 220 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo  =  A  ->  ( A (,) B )  e.  dom  vol ) )
96 xrleloe 11402 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo  <_  A  <->  ( -oo  <  A  \/ -oo  =  A ) ) )
9722, 44, 96sylancr 661 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo  <_  A  <-> 
( -oo  <  A  \/ -oo  =  A ) ) )
9846, 97mpbid 210 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo  <  A  \/ -oo  =  A ) )
9938, 95, 98mpjaod 379 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A (,) B )  e.  dom  vol )
100 ioo0 11606 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 9681 . . . . . . 7  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
102101ancoms 451 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
103100, 102bitrd 253 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  -.  A  <  B ) )
104103biimpar 483 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  =  (/) )
105104, 86syl6eqel 2498 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  e.  dom  vol )
10699, 105pm2.61dan 792 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e. 
dom  vol )
107 ndmioo 11608 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2498 . 2  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  e.  dom  vol )
109106, 108pm2.61i 164 1  |-  ( A (,) B )  e. 
dom  vol
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    \ cdif 3410    u. cun 3411    i^i cin 3412    C_ wss 3413   (/)c0 3737   {csn 3971   class class class wbr 4394   dom cdm 4822   ` cfv 5568  (class class class)co 6277   RRcr 9520   0cc0 9521   +oocpnf 9654   -oocmnf 9655   RR*cxr 9656    < clt 9657    <_ cle 9658   (,)cioo 11581   [,)cico 11583   [,]cicc 11584   vol*covol 22164   volcvol 22165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-q 11227  df-rp 11265  df-xadd 11371  df-ioo 11585  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-fl 11964  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-rlim 13459  df-sum 13656  df-xmet 18730  df-met 18731  df-ovol 22166  df-vol 22167
This theorem is referenced by:  iccmbl  22266  ovolioo  22268  ioovolcl  22269  uniioovol  22278  uniioombllem4  22285  uniioombllem5  22286  opnmblALT  22302  mbfid  22333  ditgcl  22552  ditgswap  22553  ditgsplitlem  22554  ftc1lem1  22726  ftc1lem2  22727  ftc1a  22728  ftc1lem4  22730  ftc2  22735  ftc2ditglem  22736  itgsubstlem  22739  itg2gt0cn  31423  ftc1cnnclem  31441  ftc1anclem7  31449  ftc1anclem8  31450  ftc1anc  31451  ftc2nc  31452  areacirc  31463  iocmbl  35524  cnioobibld  35525  itgpowd  35526  lhe4.4ex1a  36062  volioo  37096  itgsin0pilem1  37097  iblioosinexp  37100  itgsinexplem1  37101  itgsinexp  37102  itgcoscmulx  37117  volioc  37120  itgsincmulx  37122  iblcncfioo  37126  itgiccshift  37128  itgperiod  37129  itgsbtaddcnst  37130  wallispilem2  37197  dirkeritg  37233  fourierdlem16  37254  fourierdlem21  37259  fourierdlem22  37260  fourierdlem39  37277  fourierdlem73  37311  fourierdlem83  37321  fourierdlem103  37341  fourierdlem104  37342  fourierdlem111  37349  fourierdlem112  37350  sqwvfoura  37360  sqwvfourb  37361  etransclem18  37384  etransclem23  37389
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