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Theorem ioombl 21068
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 11432 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
213expa 1187 . . . . . . . 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 11371 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  ( A [,) B
) )
543expa 1187 . . . . . . . . . 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 4039 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  C_  ( A [,) B ) )
8 iccid 11366 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
98ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( A [,] A )  =  { A } )
109ineq1d 3572 . . . . . . . . 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 11328 . . . . . . . . . . 11  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
14 df-ioo 11325 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
15 xrltnle 9464 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
1613, 14, 15ixxdisj 11336 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( A [,] A
)  i^i  ( A (,) B ) )  =  (/) )
1711, 11, 12, 16syl3anc 1218 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( ( A [,] A )  i^i  ( A (,) B
) )  =  (/) )
1810, 17eqtr3d 2477 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  ( { A }  i^i  ( A (,) B ) )  =  (/) )
19 uneqdifeq 3788 . . . . . . . 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 11115 . . . . . . . . . 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 11163 . . . . . . . . 9  |-  ( ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  ( -oo  <  A  /\  A  <  B ) )  ->  A  e.  RR )
2723, 11, 12, 24, 25, 26syl32anc 1226 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  A  e.  RR )
28 icombl 21067 . . . . . . . 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 4039 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\ -oo  <  A )
)  ->  { A }  C_  RR )
31 ovolsn 21000 . . . . . . . . 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 21039 . . . . . . . 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 21046 . . . . . . 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 2518 . . . . 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 3521 . . . . . . . . 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 11113 . . . . . . . . . . 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 11134 . . . . . . . . . . . 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 11155 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  -> -oo  <  B )
49 pnfge 11131 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  B  <_ +oo )
5041, 49syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_ +oo )
51 df-ico 11327 . . . . . . . . . . 11  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
52 xrlenlt 9463 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
53 xrltletr 11152 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <  B  /\  B  <_ +oo )  ->  w  < +oo ) )
54 xrltletr 11152 . . . . . . . . . . 11  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( -oo  <  B  /\  B  <_  w )  -> -oo  <  w ) )
5514, 51, 52, 14, 53, 54ixxun 11337 . . . . . . . . . 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 1226 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( -oo (,) B )  u.  ( B [,) +oo ) )  =  ( -oo (,) +oo ) )
5739, 56syl5eq 2487 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  ( -oo (,) +oo )
)
58 ioomax 11391 . . . . . . . 8  |-  ( -oo (,) +oo )  =  RR
5957, 58syl6eq 2491 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( -oo (,) B
) )  =  RR )
60 ssun1 3540 . . . . . . . . 9  |-  ( B [,) +oo )  C_  ( ( B [,) +oo )  u.  ( -oo (,) B ) )
6160, 59syl5sseq 3425 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,) +oo )  C_  RR )
62 incom 3564 . . . . . . . . 9  |-  ( ( B [,) +oo )  i^i  ( -oo (,) B
) )  =  ( ( -oo (,) B
)  i^i  ( B [,) +oo ) )
6314, 51, 52ixxdisj 11336 . . . . . . . . . 10  |-  ( ( -oo  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo (,) B )  i^i  ( B [,) +oo ) )  =  (/) )
6440, 41, 43, 63syl3anc 1218 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( -oo (,) B )  i^i  ( B [,) +oo ) )  =  (/) )
6562, 64syl5eq 2487 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  i^i  ( -oo (,) B
) )  =  (/) )
66 uneqdifeq 3788 . . . . . . . 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 21044 . . . . . . 7  |-  RR  e.  dom  vol
70 xrleloe 11142 . . . . . . . . . . 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 11163 . . . . . . . . . . . 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 1310 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
7675orim1d 835 . . . . . . . . 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 21066 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B [,) +oo )  e. 
dom  vol )
79 oveq1 6119 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( B [,) +oo )  =  ( +oo [,) +oo ) )
80 pnfge 11131 . . . . . . . . . . . . 13  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
8142, 80ax-mp 5 . . . . . . . . . . . 12  |- +oo  <_ +oo
82 ico0 11367 . . . . . . . . . . . . 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 2491 . . . . . . . . . 10  |-  ( B  = +oo  ->  ( B [,) +oo )  =  (/) )
86 0mbl 21043 . . . . . . . . . 10  |-  (/)  e.  dom  vol
8785, 86syl6eqel 2531 . . . . . . . . 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 21046 . . . . . . 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 2518 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  ( -oo (,) B )  e.  dom  vol )
93 oveq1 6119 . . . . . 6  |-  ( -oo  =  A  ->  ( -oo (,) B )  =  ( A (,) B ) )
9493eleq1d 2509 . . . . 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 11142 . . . . . 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 11346 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
101 xrlenlt 9463 . . . . . . 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 2531 . . 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 11348 . . 3  |-  ( -.  ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B
)  =  (/) )
108107, 86syl6eqel 2531 . 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 965    = wceq 1369    e. wcel 1756    \ cdif 3346    u. cun 3347    i^i cin 3348    C_ wss 3349   (/)c0 3658   {csn 3898   class class class wbr 4313   dom cdm 4861   ` cfv 5439  (class class class)co 6112   RRcr 9302   0cc0 9303   +oocpnf 9436   -oocmnf 9437   RR*cxr 9438    < clt 9439    <_ cle 9440   (,)cioo 11321   [,)cico 11323   [,]cicc 11324   vol*covol 20968   volcvol 20969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-xadd 11111  df-ioo 11325  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-rlim 12988  df-sum 13185  df-xmet 17832  df-met 17833  df-ovol 20970  df-vol 20971
This theorem is referenced by:  iccmbl  21069  ovolioo  21071  ioovolcl  21072  uniioovol  21081  uniioombllem4  21088  uniioombllem5  21089  opnmblALT  21105  mbfid  21136  ditgcl  21355  ditgswap  21356  ditgsplitlem  21357  ftc1lem1  21529  ftc1lem2  21530  ftc1a  21531  ftc1lem4  21533  ftc2  21538  ftc2ditglem  21539  itgsubstlem  21542  itg2gt0cn  28473  ftc1cnnclem  28491  ftc1anclem7  28499  ftc1anclem8  28500  ftc1anc  28501  ftc2nc  28502  areacirc  28515  iocmbl  29614  cnioobibld  29615  itgpowd  29616  lhe4.4ex1a  29629  volioo  29815  itgsin0pilem1  29816  iblioosinexp  29819  itgsinexplem1  29820  itgsinexp  29821  wallispilem2  29887
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