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Theorem ioombl 21738
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 11646 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
213expa 1196 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B
) )
32adantrr 716 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
4 lbico1 11579 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  ( A [,) B
) )
543expa 1196 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
65adantrr 716 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  e.  ( A [,) B ) )
76snssd 4172 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 11574 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3699 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  ( { A }  i^i  ( A (,) B ) ) )
11 simpll 753 . . . . . . . . . 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 11536 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 11533 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 9653 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 11544 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( A [,] A
)  i^i  ( A (,) B ) )  =  (/) )
1711, 11, 12, 16syl3anc 1228 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  (/) )
1810, 17eqtr3d 2510 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3915 . . . . . . . 8  |-  ( ( { A }  C_  ( A [,) B )  /\  ( { A }  i^i  ( A (,) B ) )  =  (/) )  ->  ( ( { A }  u.  ( A (,) B ) )  =  ( A [,) B )  <->  ( ( A [,) B )  \  { A } )  =  ( A (,) B
) ) )
207, 18, 19syl2anc 661 . . . . . . 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 11323 . . . . . . . . . 10  |- -oo  e.  RR*
2322a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  -> -oo  e.  RR* )
24 simprr 756 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  -> -oo  <  A
)
25 simprl 755 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  <  B )
26 xrre2 11371 . . . . . . . . 9  |-  ( ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  ( -oo  <  A  /\  A  <  B ) )  ->  A  e.  RR )
2723, 11, 12, 24, 25, 26syl32anc 1236 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  e.  RR )
28 icombl 21737 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  e.  dom  vol )
2927, 12, 28syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( A [,) B )  e.  dom  vol )
3027snssd 4172 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 21669 . . . . . . . . 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 21709 . . . . . . . 8  |-  ( ( { A }  C_  RR  /\  ( vol* `  { A } )  =  0 )  ->  { A }  e.  dom  vol )
3430, 32, 33syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  e.  dom  vol )
35 difmbl 21716 . . . . . . 7  |-  ( ( ( A [,) B
)  e.  dom  vol  /\ 
{ A }  e.  dom  vol )  ->  (
( A [,) B
)  \  { A } )  e.  dom  vol )
3629, 34, 35syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,) B )  \  { A } )  e. 
dom  vol )
3721, 36eqeltrrd 2556 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( A (,) B )  e.  dom  vol )
3837expr 615 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo  <  A  ->  ( A (,) B )  e.  dom  vol ) )
39 uncom 3648 . . . . . . . . 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 11321 . . . . . . . . . . 11  |- +oo  e.  RR*
4342a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  -> +oo  e.  RR* )
44 simpll 753 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
45 mnfle 11342 . . . . . . . . . . . 12  |-  ( A  e.  RR*  -> -oo  <_  A )
4645ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  -> -oo  <_  A )
47 simpr 461 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
4840, 44, 41, 46, 47xrlelttrd 11363 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  -> -oo  <  B )
49 pnfge 11339 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_ +oo )
5041, 49syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_ +oo )
51 df-ico 11535 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 9652 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 11360 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_ +oo )  ->  w  < +oo ) )
54 xrltletr 11360 . . . . . . . . . . 11  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  w )  -> -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 11545 . . . . . . . . . 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 1236 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( -oo (,) B )  u.  ( B [,) +oo ) )  =  ( -oo (,) +oo ) )
5739, 56syl5eq 2520 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  ( -oo (,) +oo )
)
58 ioomax 11599 . . . . . . . 8  |-  ( -oo (,) +oo )  =  RR
5957, 58syl6eq 2524 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  RR )
60 ssun1 3667 . . . . . . . . 9  |-  ( B [,) +oo )  C_  ( ( B [,) +oo )  u.  ( -oo (,) B ) )
6160, 59syl5sseq 3552 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,) +oo )  C_  RR )
62 incom 3691 . . . . . . . . 9  |-  ( ( B [,) +oo )  i^i  ( -oo (,) B
) )  =  ( ( -oo (,) B
)  i^i  ( B [,) +oo ) )
6314, 51, 52ixxdisj 11544 . . . . . . . . . 10  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo (,) B )  i^i  ( B [,) +oo ) )  =  (/) )
6440, 41, 43, 63syl3anc 1228 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( -oo (,) B )  i^i  ( B [,) +oo ) )  =  (/) )
6562, 64syl5eq 2520 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  i^i  ( -oo (,) B
) )  =  (/) )
66 uneqdifeq 3915 . . . . . . . 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 661 . . . . . . 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 21714 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 11350 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\ +oo  e.  RR* )  ->  ( B  <_ +oo  <->  ( B  < +oo  \/  B  = +oo ) ) )
7141, 42, 70sylancl 662 . . . . . . . . . 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 11371 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  /\  ( A  <  B  /\  B  < +oo ) )  ->  B  e.  RR )
7473expr 615 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
7542, 74mp3anl3 1320 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
7675orim1d 837 . . . . . . . . 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 21736 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,) +oo )  e. 
dom  vol )
79 oveq1 6291 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( B [,) +oo )  =  ( +oo [,) +oo ) )
80 pnfge 11339 . . . . . . . . . . . . 13  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
8142, 80ax-mp 5 . . . . . . . . . . . 12  |- +oo  <_ +oo
82 ico0 11575 . . . . . . . . . . . . 13  |-  ( ( +oo  e.  RR*  /\ +oo  e.  RR* )  ->  (
( +oo [,) +oo )  =  (/)  <-> +oo  <_ +oo )
)
8342, 42, 82mp2an 672 . . . . . . . . . . . 12  |-  ( ( +oo [,) +oo )  =  (/)  <-> +oo  <_ +oo )
8481, 83mpbir 209 . . . . . . . . . . 11  |-  ( +oo [,) +oo )  =  (/)
8579, 84syl6eq 2524 . . . . . . . . . 10  |-  ( B  = +oo  ->  ( B [,) +oo )  =  (/) )
86 0mbl 21713 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2563 . . . . . . . . 9  |-  ( B  = +oo  ->  ( B [,) +oo )  e. 
dom  vol )
8878, 87jaoi 379 . . . . . . . 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 21716 . . . . . . 7  |-  ( ( RR  e.  dom  vol  /\  ( B [,) +oo )  e.  dom  vol )  ->  ( RR  \  ( B [,) +oo ) )  e.  dom  vol )
9169, 89, 90sylancr 663 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( RR  \ 
( B [,) +oo ) )  e.  dom  vol )
9268, 91eqeltrrd 2556 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo (,) B )  e.  dom  vol )
93 oveq1 6291 . . . . . 6  |-  ( -oo  =  A  ->  ( -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2536 . . . . 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 11350 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo  <_  A  <->  ( -oo  <  A  \/ -oo  =  A ) ) )
9722, 44, 96sylancr 663 . . . . 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 381 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A (,) B )  e.  dom  vol )
100 ioo0 11554 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 9652 . . . . . . 7  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
102101ancoms 453 . . . . . 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 485 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  =  (/) )
105104, 86syl6eqel 2563 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  e.  dom  vol )
10699, 105pm2.61dan 789 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e. 
dom  vol )
107 ndmioo 11556 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2563 . 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 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447   dom cdm 4999   ` cfv 5588  (class class class)co 6284   RRcr 9491   0cc0 9492   +oocpnf 9625   -oocmnf 9626   RR*cxr 9627    < clt 9628    <_ cle 9629   (,)cioo 11529   [,)cico 11531   [,]cicc 11532   vol*covol 21637   volcvol 21638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-xadd 11319  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-xmet 18211  df-met 18212  df-ovol 21639  df-vol 21640
This theorem is referenced by:  iccmbl  21739  ovolioo  21741  ioovolcl  21742  uniioovol  21751  uniioombllem4  21758  uniioombllem5  21759  opnmblALT  21775  mbfid  21806  ditgcl  22025  ditgswap  22026  ditgsplitlem  22027  ftc1lem1  22199  ftc1lem2  22200  ftc1a  22201  ftc1lem4  22203  ftc2  22208  ftc2ditglem  22209  itgsubstlem  22212  itg2gt0cn  29675  ftc1cnnclem  29693  ftc1anclem7  29701  ftc1anclem8  29702  ftc1anc  29703  ftc2nc  29704  areacirc  29717  iocmbl  30813  cnioobibld  30814  itgpowd  30815  lhe4.4ex1a  30862  volioo  31294  itgsin0pilem1  31295  iblioosinexp  31298  itgsinexplem1  31299  itgsinexp  31300  itgcoscmulx  31315  volioc  31318  itgsincmulx  31320  iblcncfioo  31324  itgiccshift  31326  itgperiod  31327  itgsbtaddcnst  31328  wallispilem2  31394  dirkeritg  31430  fourierdlem16  31451  fourierdlem21  31456  fourierdlem22  31457  fourierdlem39  31474  fourierdlem73  31508  fourierdlem83  31518  fourierdlem103  31538  fourierdlem104  31539  fourierdlem111  31546  fourierdlem112  31547  sqwvfoura  31557  sqwvfourb  31558
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