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Theorem volivth 22142
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 753 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  A  e.  dom  vol )
2 mnfxr 11348 . . . . . 6  |- -oo  e.  RR*
32a1i 11 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  -> -oo  e.  RR* )
4 iccssxr 11632 . . . . . . 7  |-  ( 0 [,] ( vol `  A
) )  C_  RR*
5 simpr 461 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  ( 0 [,] ( vol `  A
) ) )
64, 5sseldi 3497 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  RR* )
76adantr 465 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR* )
8 iccssxr 11632 . . . . . . . 8  |-  ( 0 [,] +oo )  C_  RR*
9 volf 22066 . . . . . . . . 9  |-  vol : dom  vol --> ( 0 [,] +oo )
109ffvelrni 6031 . . . . . . . 8  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  ( 0 [,] +oo ) )
118, 10sseldi 3497 . . . . . . 7  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  RR* )
1211adantr 465 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( vol `  A
)  e.  RR* )
1312adantr 465 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  ( vol `  A
)  e.  RR* )
14 0xr 9657 . . . . . . . . . 10  |-  0  e.  RR*
15 elicc1 11598 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  e.  ( 0 [,] ( vol `  A
) )  <->  ( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A ) ) ) )
1614, 12, 15sylancr 663 . . . . . . . . 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 210 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A
) ) )
1817simp2d 1009 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
0  <_  B )
1918adantr 465 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  0  <_  B
)
20 mnflt0 11359 . . . . . . . 8  |- -oo  <  0
21 xrltletr 11385 . . . . . . . 8  |-  ( ( -oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
( -oo  <  0  /\  0  <_  B )  -> -oo  <  B ) )
2220, 21mpani 676 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
0  <_  B  -> -oo 
<  B ) )
232, 14, 22mp3an12 1314 . . . . . 6  |-  ( B  e.  RR*  ->  ( 0  <_  B  -> -oo  <  B ) )
247, 19, 23sylc 60 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  -> -oo  <  B )
25 simpr 461 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  <  ( vol `  A ) )
26 xrre2 11396 . . . . 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 1236 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR )
28 volsup2 22140 . . . 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 1228 . . 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 10563 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  RR )
3130ad2antrl 727 . . . . . 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 10007 . . . . 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 465 . . . . 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 9614 . . . . . 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 10585 . . . . . . . 8  |-  ( n  e.  NN  ->  0  <  n )
3635ad2antrl 727 . . . . . . 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 10144 . . . . . . 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 210 . . . . . 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 9760 . . . . 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 11631 . . . . . 6  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  C_  RR )
4132, 31, 40syl2anc 661 . . . . 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 9566 . . . . . . 7  |-  RR  C_  CC
43 ssid 3518 . . . . . . 7  |-  CC  C_  CC
44 cncfss 21529 . . . . . . 7  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR -cn-> RR )  C_  ( RR -cn-> CC ) )
4542, 43, 44mp2an 672 . . . . . 6  |-  ( RR
-cn-> RR )  C_  ( RR -cn-> CC )
461adantr 465 . . . . . . 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 2457 . . . . . . . 8  |-  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  =  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )
4847volcn 22141 . . . . . . 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 661 . . . . . 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 3497 . . . . 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 3499 . . . . . 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 21523 . . . . . . . 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 16 . . . . . . 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 6032 . . . . . 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 470 . . . . 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 6304 . . . . . . . . . . . 12  |-  ( y  =  -u n  ->  ( -u n [,] y )  =  ( -u n [,] -u n ) )
5756ineq2d 3696 . . . . . . . . . . 11  |-  ( y  =  -u n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] -u n
) ) )
5857fveq2d 5876 . . . . . . . . . 10  |-  ( y  =  -u n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] -u n ) ) ) )
59 fvex 5882 . . . . . . . . . 10  |-  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) )  e. 
_V
6058, 47, 59fvmpt 5956 . . . . . . . . 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 16 . . . . . . . 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 3715 . . . . . . . . . . . 12  |-  ( A  i^i  ( -u n [,] -u n ) ) 
C_  ( -u n [,] -u n )
6332rexrd 9660 . . . . . . . . . . . . 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 11599 . . . . . . . . . . . . 13  |-  ( -u n  e.  RR*  ->  ( -u n [,] -u n
)  =  { -u n } )
6563, 64syl 16 . . . . . . . . . . . 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 3547 . . . . . . . . . . 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 4177 . . . . . . . . . . 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 3509 . . . . . . . . . 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 22032 . . . . . . . . . . . 12  |-  ( -u n  e.  RR  ->  ( vol* `  { -u n } )  =  0 )
7032, 69syl 16 . . . . . . . . . . 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 22024 . . . . . . . . . . 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 1228 . . . . . . . . . 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 22072 . . . . . . . . . 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 661 . . . . . . . . 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 22067 . . . . . . . . 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 16 . . . . . . . 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 2502 . . . . . . 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 465 . . . . . . 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 4476 . . . . . 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 757 . . . . . . . 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 465 . . . . . . . . 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 22102 . . . . . . . . . . . 12  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  e.  dom  vol )
8332, 31, 82syl2anc 661 . . . . . . . . . . 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 22078 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] n )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] n ) )  e. 
dom  vol )
8546, 83, 84syl2anc 661 . . . . . . . . . 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 6031 . . . . . . . . . . 11  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  ( 0 [,] +oo ) )
878, 86sseldi 3497 . . . . . . . . . 10  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  RR* )
8885, 87syl 16 . . . . . . . . 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 11380 . . . . . . . . 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 661 . . . . . . . 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 6304 . . . . . . . . . . 11  |-  ( y  =  n  ->  ( -u n [,] y )  =  ( -u n [,] n ) )
9392ineq2d 3696 . . . . . . . . . 10  |-  ( y  =  n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] n ) ) )
9493fveq2d 5876 . . . . . . . . 9  |-  ( y  =  n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
95 fvex 5882 . . . . . . . . 9  |-  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  _V
9694, 47, 95fvmpt 5956 . . . . . . . 8  |-  ( n  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
9731, 96syl 16 . . . . . . 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 4482 . . . . . 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 532 . . . . 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 21994 . . . 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 3499 . . . . . . . 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 6304 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( -u n [,] y )  =  ( -u n [,] z ) )
103102ineq2d 3696 . . . . . . . . . 10  |-  ( y  =  z  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] z ) ) )
104103fveq2d 5876 . . . . . . . . 9  |-  ( y  =  z  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
105 fvex 5882 . . . . . . . . 9  |-  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  e.  _V
106104, 47, 105fvmpt 5956 . . . . . . . 8  |-  ( z  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
107101, 106syl 16 . . . . . . 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 2459 . . . . . 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 465 . . . . . . . . 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 465 . . . . . . . . . 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 716 . . . . . . . . . 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 22102 . . . . . . . . . 10  |-  ( (
-u n  e.  RR  /\  z  e.  RR )  ->  ( -u n [,] z )  e.  dom  vol )
113110, 111, 112syl2anc 661 . . . . . . . . 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 22078 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] z )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] z ) )  e. 
dom  vol )
115109, 113, 114syl2anc 661 . . . . . . . 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 3714 . . . . . . . . 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 757 . . . . . . . 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 3520 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
x  C_  A  <->  ( A  i^i  ( -u n [,] z ) )  C_  A ) )
120 fveq2 5872 . . . . . . . . . . 11  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  ( vol `  x )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
121120eqeq1d 2459 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
( vol `  x
)  =  B  <->  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )
122119, 121anbi12d 710 . . . . . . . . 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 3210 . . . . . . . 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 1226 . . . . . . 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 615 . . . . . 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 215 . . . . 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 2949 . . . 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 2953 . 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 753 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  e.  dom  vol )
131 ssid 3518 . . . 4  |-  A  C_  A
132131a1i 11 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  C_  A
)
133 simpr 461 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  B  =  ( vol `  A ) )
134133eqcomd 2465 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  ( vol `  A )  =  B )
135 sseq1 3520 . . . . 5  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
136 fveq2 5872 . . . . . 6  |-  ( x  =  A  ->  ( vol `  x )  =  ( vol `  A
) )
137136eqeq1d 2459 . . . . 5  |-  ( x  =  A  ->  (
( vol `  x
)  =  B  <->  ( vol `  A )  =  B ) )
138135, 137anbi12d 710 . . . 4  |-  ( x  =  A  ->  (
( x  C_  A  /\  ( vol `  x
)  =  B )  <-> 
( A  C_  A  /\  ( vol `  A
)  =  B ) ) )
139138rspcev 3210 . . 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 1226 . 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 1010 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  <_  ( vol `  A
) )
142 xrleloe 11375 . . . 4  |-  ( ( B  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  <_  ( vol `  A
)  <->  ( B  < 
( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
1436, 12, 142syl2anc 661 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <_  ( vol `  A )  <->  ( B  <  ( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
144141, 143mpbid 210 . 2  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <  ( vol `  A )  \/  B  =  ( vol `  A ) ) )
145129, 140, 144mpjaodan 786 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   E.wrex 2808    i^i cin 3470    C_ wss 3471   {csn 4032   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646   -ucneg 9825   NNcn 10556   [,]cicc 11557   -cn->ccncf 21506   vol*covol 22000   volcvol 22001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-rest 14840  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-cmp 20014  df-cncf 21508  df-ovol 22002  df-vol 22003
This theorem is referenced by:  itg2const2  22274
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