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Theorem volivth 18956
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
Dummy variables  u  n  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem volivth
StepHypRef Expression
1 simpll 732 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  A  e.  dom  vol )
2 mnfxr 10451 . . . . . 6  |-  -oo  e.  RR*
32a1i 12 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  -oo  e.  RR* )
4 iccssxr 10726 . . . . . . 7  |-  ( 0 [,] ( vol `  A
) )  C_  RR*
5 simpr 449 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  ( 0 [,] ( vol `  A
) ) )
64, 5sseldi 3179 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  RR* )
76adantr 453 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR* )
8 iccssxr 10726 . . . . . . . 8  |-  ( 0 [,]  +oo )  C_  RR*
9 volf 18882 . . . . . . . . 9  |-  vol : dom  vol --> ( 0 [,] 
+oo )
109ffvelrni 5625 . . . . . . . 8  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  ( 0 [,] 
+oo ) )
118, 10sseldi 3179 . . . . . . 7  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  RR* )
1211adantr 453 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( vol `  A
)  e.  RR* )
1312adantr 453 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  ( vol `  A
)  e.  RR* )
14 0xr 8873 . . . . . . . . . 10  |-  0  e.  RR*
15 elicc1 10694 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  e.  ( 0 [,] ( vol `  A
) )  <->  ( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A ) ) ) )
1614, 12, 15sylancr 646 . . . . . . . . 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 203 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A
) ) )
1817simp2d 970 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
0  <_  B )
1918adantr 453 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  0  <_  B
)
20 0re 8833 . . . . . . . . 9  |-  0  e.  RR
21 mnflt 10459 . . . . . . . . 9  |-  ( 0  e.  RR  ->  -oo  <  0 )
2220, 21ax-mp 10 . . . . . . . 8  |-  -oo  <  0
23 xrltletr 10483 . . . . . . . 8  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
(  -oo  <  0  /\  0  <_  B )  ->  -oo  <  B ) )
2422, 23mpani 659 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
0  <_  B  ->  -oo 
<  B ) )
252, 14, 24mp3an12 1269 . . . . . 6  |-  ( B  e.  RR*  ->  ( 0  <_  B  ->  -oo  <  B ) )
267, 19, 25sylc 58 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  -oo  <  B )
27 simpr 449 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  <  ( vol `  A ) )
28 xrre2 10494 . . . . 5  |-  ( ( (  -oo  e.  RR*  /\  B  e.  RR*  /\  ( vol `  A )  e. 
RR* )  /\  (  -oo  <  B  /\  B  <  ( vol `  A
) ) )  ->  B  e.  RR )
293, 7, 13, 26, 27, 28syl32anc 1192 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR )
30 volsup2 18954 . . . 4  |-  ( ( A  e.  dom  vol  /\  B  e.  RR  /\  B  <  ( vol `  A
) )  ->  E. n  e.  NN  B  <  ( vol `  ( A  i^i  ( -u n [,] n
) ) ) )
311, 29, 27, 30syl3anc 1184 . . 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 ) ) ) )
32 nnre 9748 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  RR )
3332ad2antrl 710 . . . . . . . 8  |-  ( ( ( ( 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 )
3433renegcld 9205 . . . . . . 7  |-  ( ( ( ( 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 )
3529adantr 453 . . . . . . 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  e.  RR )
3620a1i 12 . . . . . . . 8  |-  ( ( ( ( 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 )
37 nngt0 9770 . . . . . . . . . 10  |-  ( n  e.  NN  ->  0  <  n )
3837ad2antrl 710 . . . . . . . . 9  |-  ( ( ( ( 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 )
3933lt0neg2d 9338 . . . . . . . . 9  |-  ( ( ( ( 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 ) )
4038, 39mpbid 203 . . . . . . . 8  |-  ( ( ( ( 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 )
4134, 36, 33, 40, 38lttrd 8972 . . . . . . 7  |-  ( ( ( ( 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 )
42 iccssre 10725 . . . . . . . 8  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  C_  RR )
4334, 33, 42syl2anc 644 . . . . . . 7  |-  ( ( ( ( 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 )
44 ax-resscn 8789 . . . . . . . . 9  |-  RR  C_  CC
45 ssid 3198 . . . . . . . . 9  |-  CC  C_  CC
46 cncfss 18397 . . . . . . . . 9  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR -cn-> RR )  C_  ( RR -cn-> CC ) )
4744, 45, 46mp2an 655 . . . . . . . 8  |-  ( RR
-cn-> RR )  C_  ( RR -cn-> CC )
481adantr 453 . . . . . . . . 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  e.  dom  vol )
49 eqid 2284 . . . . . . . . . 10  |-  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  =  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )
5049volcn 18955 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  -u n  e.  RR )  ->  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  e.  ( RR
-cn-> RR ) )
5148, 34, 50syl2anc 644 . . . . . . . 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 ) ) ) )  e.  ( RR
-cn-> RR ) )
5247, 51sseldi 3179 . . . . . . 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 ) ) ) )  e.  ( RR
-cn-> CC ) )
5343sselda 3181 . . . . . . . 8  |-  ( ( ( ( ( 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 )
54 cncff 18391 . . . . . . . . . 10  |-  ( ( 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 )
5551, 54syl 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 ) ) ) ) )  ->  (
y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) : RR --> RR )
56 ffvelrn 5624 . . . . . . . . 9  |-  ( ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) : RR --> RR  /\  u  e.  RR )  ->  ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  u )  e.  RR )
5755, 56sylan 459 . . . . . . . 8  |-  ( ( ( ( ( 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 )
5853, 57syldan 458 . . . . . . 7  |-  ( ( ( ( ( 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 )
59 oveq2 5827 . . . . . . . . . . . . . 14  |-  ( y  =  -u n  ->  ( -u n [,] y )  =  ( -u n [,] -u n ) )
6059ineq2d 3371 . . . . . . . . . . . . 13  |-  ( y  =  -u n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] -u n
) ) )
6160fveq2d 5489 . . . . . . . . . . . 12  |-  ( y  =  -u n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] -u n ) ) ) )
62 fvex 5499 . . . . . . . . . . . 12  |-  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) )  e. 
_V
6361, 49, 62fvmpt 5563 . . . . . . . . . . 11  |-  ( -u n  e.  RR  ->  ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  -u n
)  =  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) ) )
6434, 63syl 17 . . . . . . . . . 10  |-  ( ( ( ( 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
) ) ) )
65 inss2 3391 . . . . . . . . . . . . . 14  |-  ( A  i^i  ( -u n [,] -u n ) ) 
C_  ( -u n [,] -u n )
6634rexrd 8876 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 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* )
67 iccid 10695 . . . . . . . . . . . . . . 15  |-  ( -u n  e.  RR*  ->  ( -u n [,] -u n
)  =  { -u n } )
6866, 67syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 } )
6965, 68syl5sseq 3227 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 } )
7034snssd 3761 . . . . . . . . . . . . 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 }  C_  RR )
7169, 70sstrd 3190 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )
72 ovolsn 18848 . . . . . . . . . . . . . 14  |-  ( -u n  e.  RR  ->  ( vol * `  { -u n } )  =  0 )
7334, 72syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )
74 ovolssnul 18840 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
7569, 70, 73, 74syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )
76 nulmbl 18887 . . . . . . . . . . . 12  |-  ( ( ( 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 )
7771, 75, 76syl2anc 644 . . . . . . . . . . 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 ) )  e.  dom  vol )
78 mblvol 18883 . . . . . . . . . . 11  |-  ( ( 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 ) ) ) )
7977, 78syl 17 . . . . . . . . . 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
) ) )  =  ( vol * `  ( A  i^i  ( -u n [,] -u n
) ) ) )
8064, 79, 753eqtrd 2320 . . . . . . . . 9  |-  ( ( ( ( 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 )
8119adantr 453 . . . . . . . . 9  |-  ( ( ( ( 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 )
8280, 81eqbrtrd 4044 . . . . . . . 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
)  <_  B )
83 simprr 735 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) ) )
847adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( 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* )
85 iccmbl 18917 . . . . . . . . . . . . . 14  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  e.  dom  vol )
8634, 33, 85syl2anc 644 . . . . . . . . . . . . 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 [,] n )  e.  dom  vol )
87 inmbl 18893 . . . . . . . . . . . . 13  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] n )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] n ) )  e. 
dom  vol )
8848, 86, 87syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )
899ffvelrni 5625 . . . . . . . . . . . . 13  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  ( 0 [,] 
+oo ) )
908, 89sseldi 3179 . . . . . . . . . . . 12  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  RR* )
9188, 90syl 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 `  ( A  i^i  ( -u n [,] n
) ) )  e. 
RR* )
92 xrltle 10478 . . . . . . . . . . 11  |-  ( ( 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
) ) ) ) )
9384, 91, 92syl2anc 644 . . . . . . . . . 10  |-  ( ( ( ( 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
) ) ) ) )
9483, 93mpd 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 ) ) ) ) )  ->  B  <_  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
95 oveq2 5827 . . . . . . . . . . . . 13  |-  ( y  =  n  ->  ( -u n [,] y )  =  ( -u n [,] n ) )
9695ineq2d 3371 . . . . . . . . . . . 12  |-  ( y  =  n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] n ) ) )
9796fveq2d 5489 . . . . . . . . . . 11  |-  ( y  =  n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
98 fvex 5499 . . . . . . . . . . 11  |-  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  _V
9997, 49, 98fvmpt 5563 . . . . . . . . . 10  |-  ( n  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
10033, 99syl 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 ) ) ) ) )  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
10194, 100breqtrrd 4050 . . . . . . . 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  <_  ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n ) )
10282, 101jca 520 . . . . . . 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
)  <_  B  /\  B  <_  ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n ) ) )
10334, 33, 35, 41, 43, 52, 58, 102ivthle 18810 . . . . . 6  |-  ( ( ( ( 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 )
10443sselda 3181 . . . . . . . . . 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 ) )  -> 
z  e.  RR )
105 oveq2 5827 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  ( -u n [,] y )  =  ( -u n [,] z ) )
106105ineq2d 3371 . . . . . . . . . . . 12  |-  ( y  =  z  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] z ) ) )
107106fveq2d 5489 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
108 fvex 5499 . . . . . . . . . . 11  |-  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  e.  _V
109107, 49, 108fvmpt 5563 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
110104, 109syl 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 ) ) ) ) )  /\  z  e.  ( -u n [,] n ) )  -> 
( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
111110eqeq1d 2292 . . . . . . . 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 ) )  -> 
( ( ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  B  <-> 
( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )
11248adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 )
11334adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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 )
114104adantrr 699 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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 )
115 iccmbl 18917 . . . . . . . . . . . 12  |-  ( (
-u n  e.  RR  /\  z  e.  RR )  ->  ( -u n [,] z )  e.  dom  vol )
116113, 114, 115syl2anc 644 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 )
117 inmbl 18893 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] z )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] z ) )  e. 
dom  vol )
118112, 116, 117syl2anc 644 . . . . . . . . . 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 ) )  ->  ( A  i^i  ( -u n [,] z
) )  e.  dom  vol )
119 inss1 3390 . . . . . . . . . . 11  |-  ( A  i^i  ( -u n [,] z ) )  C_  A
120119a1i 12 . . . . . . . . . 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 ) )  ->  ( A  i^i  ( -u n [,] z
) )  C_  A
)
121 simprr 735 . . . . . . . . . 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 ) )  ->  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B )
122 sseq1 3200 . . . . . . . . . . . 12  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
x  C_  A  <->  ( A  i^i  ( -u n [,] z ) )  C_  A ) )
123 fveq2 5485 . . . . . . . . . . . . 13  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  ( vol `  x )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
124123eqeq1d 2292 . . . . . . . . . . . 12  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
( vol `  x
)  =  B  <->  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )
125122, 124anbi12d 693 . . . . . . . . . . 11  |-  ( 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 ) ) )
126125rspcev 2885 . . . . . . . . . 10  |-  ( ( ( 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 ) )
127118, 120, 121, 126syl12anc 1182 . . . . . . . . 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 ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) )
128127expr 600 . . . . . . . 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  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) ) )
129111, 128sylbid 208 . . . . . . 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 )  =  B  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) ) )
130129rexlimdva 2668 . . . . . 6  |-  ( ( ( ( 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 ) ) )
131103, 130mpd 16 . . . . 5  |-  ( ( ( ( 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 ) )
132131expr 600 . . . 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. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) ) )
133132rexlimdva 2668 . . 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
) ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) ) )
13431, 133mpd 16 . 2  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) )
135 simpll 732 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  e.  dom  vol )
136 ssid 3198 . . . 4  |-  A  C_  A
137136a1i 12 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  C_  A
)
138 simpr 449 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  B  =  ( vol `  A ) )
139138eqcomd 2289 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  ( vol `  A )  =  B )
140 sseq1 3200 . . . . 5  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
141 fveq2 5485 . . . . . 6  |-  ( x  =  A  ->  ( vol `  x )  =  ( vol `  A
) )
142141eqeq1d 2292 . . . . 5  |-  ( x  =  A  ->  (
( vol `  x
)  =  B  <->  ( vol `  A )  =  B ) )
143140, 142anbi12d 693 . . . 4  |-  ( x  =  A  ->  (
( x  C_  A  /\  ( vol `  x
)  =  B )  <-> 
( A  C_  A  /\  ( vol `  A
)  =  B ) ) )
144143rspcev 2885 . . 3  |-  ( ( A  e.  dom  vol  /\  ( A  C_  A  /\  ( vol `  A
)  =  B ) )  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) )
145135, 137, 139, 144syl12anc 1182 . 2  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) )
14617simp3d 971 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  <_  ( vol `  A
) )
147 xrleloe 10473 . . . 4  |-  ( ( B  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  <_  ( vol `  A
)  <->  ( B  < 
( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
1486, 12, 147syl2anc 644 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <_  ( vol `  A )  <->  ( B  <  ( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
149146, 148mpbid 203 . 2  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <  ( vol `  A )  \/  B  =  ( vol `  A ) ) )
150134, 145, 149mpjaodan 763 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 set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   E.wrex 2545    i^i cin 3152    C_ wss 3153   {csn 3641   class class class wbr 4024    e. cmpt 4078   dom cdm 4688   -->wf 5217   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732    +oocpnf 8859    -oocmnf 8860   RR*cxr 8861    < clt 8862    <_ cle 8863   -ucneg 9033   NNcn 9741   [,]cicc 10653   -cn->ccncf 18374   vol *covol 18816   volcvol 18817
This theorem is referenced by:  itg2const2  19090
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cc 8056  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-disj 3995  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-rlim 11957  df-sum 12153  df-rest 13321  df-topgen 13338  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-top 16630  df-bases 16632  df-topon 16633  df-cmp 17108  df-cncf 18376  df-ovol 18818  df-vol 18819
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