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Theorem ioombl 21953
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 11657 . . . . . . . . 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 716 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
4 lbico1 11590 . . . . . . . . . . 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 716 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  e.  ( A [,) B ) )
76snssd 4160 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 11585 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3684 . . . . . . . . 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 755 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  B  e.  RR* )
13 df-icc 11547 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 11544 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 9656 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 11555 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( A [,] A
)  i^i  ( A (,) B ) )  =  (/) )
1711, 11, 12, 16syl3anc 1229 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  (/) )
1810, 17eqtr3d 2486 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3902 . . . . . . . 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 11334 . . . . . . . . . 10  |- -oo  e.  RR*
2322a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  -> -oo  e.  RR* )
24 simprr 757 . . . . . . . . 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 11382 . . . . . . . . 9  |-  ( ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  ( -oo  <  A  /\  A  <  B ) )  ->  A  e.  RR )
2723, 11, 12, 24, 25, 26syl32anc 1237 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  e.  RR )
28 icombl 21952 . . . . . . . 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 4160 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 21884 . . . . . . . . 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 21924 . . . . . . . 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 21931 . . . . . . 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 2532 . . . . 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 3633 . . . . . . . . 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 755 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
42 pnfxr 11332 . . . . . . . . . . 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 11353 . . . . . . . . . . . 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 11374 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  -> -oo  <  B )
49 pnfge 11350 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_ +oo )
5041, 49syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_ +oo )
51 df-ico 11546 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 9655 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 11371 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_ +oo )  ->  w  < +oo ) )
54 xrltletr 11371 . . . . . . . . . . 11  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  w )  -> -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 11556 . . . . . . . . . 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 1237 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( -oo (,) B )  u.  ( B [,) +oo ) )  =  ( -oo (,) +oo ) )
5739, 56syl5eq 2496 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  ( -oo (,) +oo )
)
58 ioomax 11610 . . . . . . . 8  |-  ( -oo (,) +oo )  =  RR
5957, 58syl6eq 2500 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  RR )
60 ssun1 3652 . . . . . . . . 9  |-  ( B [,) +oo )  C_  ( ( B [,) +oo )  u.  ( -oo (,) B ) )
6160, 59syl5sseq 3537 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,) +oo )  C_  RR )
62 incom 3676 . . . . . . . . 9  |-  ( ( B [,) +oo )  i^i  ( -oo (,) B
) )  =  ( ( -oo (,) B
)  i^i  ( B [,) +oo ) )
6314, 51, 52ixxdisj 11555 . . . . . . . . . 10  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo (,) B )  i^i  ( B [,) +oo ) )  =  (/) )
6440, 41, 43, 63syl3anc 1229 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( -oo (,) B )  i^i  ( B [,) +oo ) )  =  (/) )
6562, 64syl5eq 2496 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  i^i  ( -oo (,) B
) )  =  (/) )
66 uneqdifeq 3902 . . . . . . . 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 21929 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 11361 . . . . . . . . . . 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 11382 . . . . . . . . . . . 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 1321 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
7675orim1d 839 . . . . . . . . 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 21951 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,) +oo )  e. 
dom  vol )
79 oveq1 6288 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( B [,) +oo )  =  ( +oo [,) +oo ) )
80 pnfge 11350 . . . . . . . . . . . . 13  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
8142, 80ax-mp 5 . . . . . . . . . . . 12  |- +oo  <_ +oo
82 ico0 11586 . . . . . . . . . . . . 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 2500 . . . . . . . . . 10  |-  ( B  = +oo  ->  ( B [,) +oo )  =  (/) )
86 0mbl 21928 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2539 . . . . . . . . 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 21931 . . . . . . 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 2532 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo (,) B )  e.  dom  vol )
93 oveq1 6288 . . . . . 6  |-  ( -oo  =  A  ->  ( -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2512 . . . . 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 11361 . . . . . 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 11565 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 9655 . . . . . . 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 2539 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A (,) B )  e.  dom  vol )
10699, 105pm2.61dan 791 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e. 
dom  vol )
107 ndmioo 11567 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2539 . 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 974    = wceq 1383    e. wcel 1804    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014   class class class wbr 4437   dom cdm 4989   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495   +oocpnf 9628   -oocmnf 9629   RR*cxr 9630    < clt 9631    <_ cle 9632   (,)cioo 11540   [,)cico 11542   [,]cicc 11543   vol*covol 21852   volcvol 21853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-q 11194  df-rp 11232  df-xadd 11330  df-ioo 11544  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-xmet 18391  df-met 18392  df-ovol 21854  df-vol 21855
This theorem is referenced by:  iccmbl  21954  ovolioo  21956  ioovolcl  21957  uniioovol  21966  uniioombllem4  21973  uniioombllem5  21974  opnmblALT  21990  mbfid  22021  ditgcl  22240  ditgswap  22241  ditgsplitlem  22242  ftc1lem1  22414  ftc1lem2  22415  ftc1a  22416  ftc1lem4  22418  ftc2  22423  ftc2ditglem  22424  itgsubstlem  22427  itg2gt0cn  30046  ftc1cnnclem  30064  ftc1anclem7  30072  ftc1anclem8  30073  ftc1anc  30074  ftc2nc  30075  areacirc  30088  iocmbl  31156  cnioobibld  31157  itgpowd  31158  lhe4.4ex1a  31210  volioo  31701  itgsin0pilem1  31702  iblioosinexp  31705  itgsinexplem1  31706  itgsinexp  31707  itgcoscmulx  31722  volioc  31725  itgsincmulx  31727  iblcncfioo  31731  itgiccshift  31733  itgperiod  31734  itgsbtaddcnst  31735  wallispilem2  31802  dirkeritg  31838  fourierdlem16  31859  fourierdlem21  31864  fourierdlem22  31865  fourierdlem39  31882  fourierdlem73  31916  fourierdlem83  31926  fourierdlem103  31946  fourierdlem104  31947  fourierdlem111  31954  fourierdlem112  31955  sqwvfoura  31965  sqwvfourb  31966  etransclem18  31989  etransclem23  31994
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