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Theorem volivth 22565
Description: The Intermediate Value Theorem for the Lebesgue volume function. For any positive  B  <_  ( vol `  A ), there is a measurable subset of  A whose volume is  B. (Contributed by Mario Carneiro, 30-Aug-2014.)
Assertion
Ref Expression
volivth  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem volivth
Dummy variables  u  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 760 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  A  e.  dom  vol )
2 mnfxr 11414 . . . . . 6  |- -oo  e.  RR*
32a1i 11 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  -> -oo  e.  RR* )
4 iccssxr 11717 . . . . . . 7  |-  ( 0 [,] ( vol `  A
) )  C_  RR*
5 simpr 463 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  ( 0 [,] ( vol `  A
) ) )
64, 5sseldi 3430 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  RR* )
76adantr 467 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR* )
8 iccssxr 11717 . . . . . . . 8  |-  ( 0 [,] +oo )  C_  RR*
9 volf 22483 . . . . . . . . 9  |-  vol : dom  vol --> ( 0 [,] +oo )
109ffvelrni 6021 . . . . . . . 8  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  ( 0 [,] +oo ) )
118, 10sseldi 3430 . . . . . . 7  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  RR* )
1211adantr 467 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( vol `  A
)  e.  RR* )
1312adantr 467 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  ( vol `  A
)  e.  RR* )
14 0xr 9687 . . . . . . . . . 10  |-  0  e.  RR*
15 elicc1 11680 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  e.  ( 0 [,] ( vol `  A
) )  <->  ( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A ) ) ) )
1614, 12, 15sylancr 669 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  e.  ( 0 [,] ( vol `  A ) )  <->  ( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A ) ) ) )
175, 16mpbid 214 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A
) ) )
1817simp2d 1021 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
0  <_  B )
1918adantr 467 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  0  <_  B
)
20 mnflt0 11427 . . . . . . . 8  |- -oo  <  0
21 xrltletr 11454 . . . . . . . 8  |-  ( ( -oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
( -oo  <  0  /\  0  <_  B )  -> -oo  <  B ) )
2220, 21mpani 682 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
0  <_  B  -> -oo 
<  B ) )
232, 14, 22mp3an12 1354 . . . . . 6  |-  ( B  e.  RR*  ->  ( 0  <_  B  -> -oo  <  B ) )
247, 19, 23sylc 62 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  -> -oo  <  B )
25 simpr 463 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  <  ( vol `  A ) )
26 xrre2 11465 . . . . 5  |-  ( ( ( -oo  e.  RR*  /\  B  e.  RR*  /\  ( vol `  A )  e. 
RR* )  /\  ( -oo  <  B  /\  B  <  ( vol `  A
) ) )  ->  B  e.  RR )
273, 7, 13, 24, 25, 26syl32anc 1276 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR )
28 volsup2 22563 . . . 4  |-  ( ( A  e.  dom  vol  /\  B  e.  RR  /\  B  <  ( vol `  A
) )  ->  E. n  e.  NN  B  <  ( vol `  ( A  i^i  ( -u n [,] n
) ) ) )
291, 27, 25, 28syl3anc 1268 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  E. n  e.  NN  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
30 nnre 10616 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  RR )
3130ad2antrl 734 . . . . . 6  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  n  e.  RR )
3231renegcld 10046 . . . . 5  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  -u n  e.  RR )
3327adantr 467 . . . . 5  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  B  e.  RR )
34 0red 9644 . . . . . 6  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  0  e.  RR )
35 nngt0 10638 . . . . . . . 8  |-  ( n  e.  NN  ->  0  <  n )
3635ad2antrl 734 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  0  <  n )
3731lt0neg2d 10184 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
0  <  n  <->  -u n  <  0 ) )
3836, 37mpbid 214 . . . . . 6  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  -u n  <  0 )
3932, 34, 31, 38, 36lttrd 9796 . . . . 5  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  -u n  <  n )
40 iccssre 11716 . . . . . 6  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  C_  RR )
4132, 31, 40syl2anc 667 . . . . 5  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( -u n [,] n ) 
C_  RR )
42 ax-resscn 9596 . . . . . . 7  |-  RR  C_  CC
43 ssid 3451 . . . . . . 7  |-  CC  C_  CC
44 cncfss 21931 . . . . . . 7  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR -cn-> RR )  C_  ( RR -cn-> CC ) )
4542, 43, 44mp2an 678 . . . . . 6  |-  ( RR
-cn-> RR )  C_  ( RR -cn-> CC )
461adantr 467 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  A  e.  dom  vol )
47 eqid 2451 . . . . . . . 8  |-  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  =  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )
4847volcn 22564 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  -u n  e.  RR )  ->  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  e.  ( RR
-cn-> RR ) )
4946, 32, 48syl2anc 667 . . . . . 6  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  e.  ( RR
-cn-> RR ) )
5045, 49sseldi 3430 . . . . 5  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  e.  ( RR
-cn-> CC ) )
5141sselda 3432 . . . . . 6  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  u  e.  ( -u n [,] n ) )  ->  u  e.  RR )
52 cncff 21925 . . . . . . . 8  |-  ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  e.  ( RR
-cn-> RR )  ->  (
y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) : RR --> RR )
5349, 52syl 17 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) : RR --> RR )
5453ffvelrnda 6022 . . . . . 6  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  u  e.  RR )  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  u )  e.  RR )
5551, 54syldan 473 . . . . 5  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  u  e.  ( -u n [,] n ) )  -> 
( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  u )  e.  RR )
56 oveq2 6298 . . . . . . . . . . . 12  |-  ( y  =  -u n  ->  ( -u n [,] y )  =  ( -u n [,] -u n ) )
5756ineq2d 3634 . . . . . . . . . . 11  |-  ( y  =  -u n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] -u n
) ) )
5857fveq2d 5869 . . . . . . . . . 10  |-  ( y  =  -u n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] -u n ) ) ) )
59 fvex 5875 . . . . . . . . . 10  |-  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) )  e. 
_V
6058, 47, 59fvmpt 5948 . . . . . . . . 9  |-  ( -u n  e.  RR  ->  ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  -u n
)  =  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) ) )
6132, 60syl 17 . . . . . . . 8  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  -u n
)  =  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) ) )
62 inss2 3653 . . . . . . . . . . . 12  |-  ( A  i^i  ( -u n [,] -u n ) ) 
C_  ( -u n [,] -u n )
6332rexrd 9690 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  -u n  e.  RR* )
64 iccid 11681 . . . . . . . . . . . . 13  |-  ( -u n  e.  RR*  ->  ( -u n [,] -u n
)  =  { -u n } )
6563, 64syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( -u n [,] -u n
)  =  { -u n } )
6662, 65syl5sseq 3480 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( A  i^i  ( -u n [,] -u n ) ) 
C_  { -u n } )
6732snssd 4117 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  { -u n }  C_  RR )
6866, 67sstrd 3442 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( A  i^i  ( -u n [,] -u n ) ) 
C_  RR )
69 ovolsn 22448 . . . . . . . . . . . 12  |-  ( -u n  e.  RR  ->  ( vol* `  { -u n } )  =  0 )
7032, 69syl 17 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( vol* `  { -u n } )  =  0 )
71 ovolssnul 22440 . . . . . . . . . . 11  |-  ( ( ( A  i^i  ( -u n [,] -u n
) )  C_  { -u n }  /\  { -u n }  C_  RR  /\  ( vol* `  { -u n } )  =  0 )  ->  ( vol* `  ( A  i^i  ( -u n [,] -u n ) ) )  =  0 )
7266, 67, 70, 71syl3anc 1268 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( vol* `  ( A  i^i  ( -u n [,] -u n ) ) )  =  0 )
73 nulmbl 22489 . . . . . . . . . 10  |-  ( ( ( A  i^i  ( -u n [,] -u n
) )  C_  RR  /\  ( vol* `  ( A  i^i  ( -u n [,] -u n
) ) )  =  0 )  ->  ( A  i^i  ( -u n [,] -u n ) )  e.  dom  vol )
7468, 72, 73syl2anc 667 . . . . . . . . 9  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( A  i^i  ( -u n [,] -u n ) )  e.  dom  vol )
75 mblvol 22484 . . . . . . . . 9  |-  ( ( A  i^i  ( -u n [,] -u n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] -u n ) ) )  =  ( vol* `  ( A  i^i  ( -u n [,] -u n ) ) ) )
7674, 75syl 17 . . . . . . . 8  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) )  =  ( vol* `  ( A  i^i  ( -u n [,] -u n
) ) ) )
7761, 76, 723eqtrd 2489 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  -u n
)  =  0 )
7819adantr 467 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  0  <_  B )
7977, 78eqbrtrd 4423 . . . . . 6  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  -u n
)  <_  B )
80 simprr 766 . . . . . . . 8  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
817adantr 467 . . . . . . . . 9  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  B  e.  RR* )
82 iccmbl 22519 . . . . . . . . . . . 12  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  e.  dom  vol )
8332, 31, 82syl2anc 667 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( -u n [,] n )  e.  dom  vol )
84 inmbl 22495 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] n )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] n ) )  e. 
dom  vol )
8546, 83, 84syl2anc 667 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( A  i^i  ( -u n [,] n ) )  e. 
dom  vol )
869ffvelrni 6021 . . . . . . . . . . 11  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  ( 0 [,] +oo ) )
878, 86sseldi 3430 . . . . . . . . . 10  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  RR* )
8885, 87syl 17 . . . . . . . . 9  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( vol `  ( A  i^i  ( -u n [,] n
) ) )  e. 
RR* )
89 xrltle 11448 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  ( vol `  ( A  i^i  ( -u n [,] n
) ) )  e. 
RR* )  ->  ( B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  ->  B  <_  ( vol `  ( A  i^i  ( -u n [,] n
) ) ) ) )
9081, 88, 89syl2anc 667 . . . . . . . 8  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  ->  B  <_  ( vol `  ( A  i^i  ( -u n [,] n
) ) ) ) )
9180, 90mpd 15 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  B  <_  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
92 oveq2 6298 . . . . . . . . . . 11  |-  ( y  =  n  ->  ( -u n [,] y )  =  ( -u n [,] n ) )
9392ineq2d 3634 . . . . . . . . . 10  |-  ( y  =  n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] n ) ) )
9493fveq2d 5869 . . . . . . . . 9  |-  ( y  =  n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
95 fvex 5875 . . . . . . . . 9  |-  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  _V
9694, 47, 95fvmpt 5948 . . . . . . . 8  |-  ( n  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
9731, 96syl 17 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
9891, 97breqtrrd 4429 . . . . . 6  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  B  <_  ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n ) )
9979, 98jca 535 . . . . 5  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  (
( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  -u n
)  <_  B  /\  B  <_  ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n ) ) )
10032, 31, 33, 39, 41, 50, 55, 99ivthle 22407 . . . 4  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  E. z  e.  ( -u n [,] n ) ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  B )
10141sselda 3432 . . . . . . . 8  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  z  e.  ( -u n [,] n ) )  -> 
z  e.  RR )
102 oveq2 6298 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( -u n [,] y )  =  ( -u n [,] z ) )
103102ineq2d 3634 . . . . . . . . . 10  |-  ( y  =  z  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] z ) ) )
104103fveq2d 5869 . . . . . . . . 9  |-  ( y  =  z  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
105 fvex 5875 . . . . . . . . 9  |-  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  e.  _V
106104, 47, 105fvmpt 5948 . . . . . . . 8  |-  ( z  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
107101, 106syl 17 . . . . . . 7  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  z  e.  ( -u n [,] n ) )  -> 
( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
108107eqeq1d 2453 . . . . . 6  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  z  e.  ( -u n [,] n ) )  -> 
( ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  B  <-> 
( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )
10946adantr 467 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  (
z  e.  ( -u n [,] n )  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  A  e.  dom  vol )
11032adantr 467 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  (
z  e.  ( -u n [,] n )  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  -u n  e.  RR )
111101adantrr 723 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  (
z  e.  ( -u n [,] n )  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  z  e.  RR )
112 iccmbl 22519 . . . . . . . . . 10  |-  ( (
-u n  e.  RR  /\  z  e.  RR )  ->  ( -u n [,] z )  e.  dom  vol )
113110, 111, 112syl2anc 667 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  (
z  e.  ( -u n [,] n )  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  ( -u n [,] z )  e.  dom  vol )
114 inmbl 22495 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] z )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] z ) )  e. 
dom  vol )
115109, 113, 114syl2anc 667 . . . . . . . 8  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  (
z  e.  ( -u n [,] n )  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  ( A  i^i  ( -u n [,] z
) )  e.  dom  vol )
116 inss1 3652 . . . . . . . . 9  |-  ( A  i^i  ( -u n [,] z ) )  C_  A
117116a1i 11 . . . . . . . 8  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  (
z  e.  ( -u n [,] n )  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  ( A  i^i  ( -u n [,] z
) )  C_  A
)
118 simprr 766 . . . . . . . 8  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  (
z  e.  ( -u n [,] n )  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B )
119 sseq1 3453 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
x  C_  A  <->  ( A  i^i  ( -u n [,] z ) )  C_  A ) )
120 fveq2 5865 . . . . . . . . . . 11  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  ( vol `  x )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
121120eqeq1d 2453 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
( vol `  x
)  =  B  <->  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )
122119, 121anbi12d 717 . . . . . . . . 9  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
( x  C_  A  /\  ( vol `  x
)  =  B )  <-> 
( ( A  i^i  ( -u n [,] z
) )  C_  A  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) ) )
123122rspcev 3150 . . . . . . . 8  |-  ( ( ( A  i^i  ( -u n [,] z ) )  e.  dom  vol  /\  ( ( A  i^i  ( -u n [,] z
) )  C_  A  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) )
124115, 117, 118, 123syl12anc 1266 . . . . . . 7  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  (
z  e.  ( -u n [,] n )  /\  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) )
125124expr 620 . . . . . 6  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  z  e.  ( -u n [,] n ) )  -> 
( ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) ) )
126108, 125sylbid 219 . . . . 5  |-  ( ( ( ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  /\  B  <  ( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  /\  z  e.  ( -u n [,] n ) )  -> 
( ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  B  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) ) )
127126rexlimdva 2879 . . . 4  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  ( E. z  e.  ( -u n [,] n ) ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  B  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) ) )
128100, 127mpd 15 . . 3  |-  ( ( ( ( A  e. 
dom  vol  /\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  < 
( vol `  A
) )  /\  (
n  e.  NN  /\  B  <  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) ) )  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) )
12929, 128rexlimddv 2883 . 2  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) )
130 simpll 760 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  e.  dom  vol )
131 ssid 3451 . . . 4  |-  A  C_  A
132131a1i 11 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  C_  A
)
133 simpr 463 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  B  =  ( vol `  A ) )
134133eqcomd 2457 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  ( vol `  A )  =  B )
135 sseq1 3453 . . . . 5  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
136 fveq2 5865 . . . . . 6  |-  ( x  =  A  ->  ( vol `  x )  =  ( vol `  A
) )
137136eqeq1d 2453 . . . . 5  |-  ( x  =  A  ->  (
( vol `  x
)  =  B  <->  ( vol `  A )  =  B ) )
138135, 137anbi12d 717 . . . 4  |-  ( x  =  A  ->  (
( x  C_  A  /\  ( vol `  x
)  =  B )  <-> 
( A  C_  A  /\  ( vol `  A
)  =  B ) ) )
139138rspcev 3150 . . 3  |-  ( ( A  e.  dom  vol  /\  ( A  C_  A  /\  ( vol `  A
)  =  B ) )  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) )
140130, 132, 134, 139syl12anc 1266 . 2  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) )
14117simp3d 1022 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  <_  ( vol `  A
) )
142 xrleloe 11443 . . . 4  |-  ( ( B  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  <_  ( vol `  A
)  <->  ( B  < 
( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
1436, 12, 142syl2anc 667 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <_  ( vol `  A )  <->  ( B  <  ( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
144141, 143mpbid 214 . 2  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <  ( vol `  A )  \/  B  =  ( vol `  A ) ) )
145129, 140, 144mpjaodan 795 1  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   E.wrex 2738    i^i cin 3403    C_ wss 3404   {csn 3968   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   +oocpnf 9672   -oocmnf 9673   RR*cxr 9674    < clt 9675    <_ cle 9676   -ucneg 9861   NNcn 10609   [,]cicc 11638   -cn->ccncf 21908   vol*covol 22413   volcvol 22415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-rest 15321  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-top 19921  df-bases 19922  df-topon 19923  df-cmp 20402  df-cncf 21910  df-ovol 22416  df-vol 22418
This theorem is referenced by:  itg2const2  22699
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