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Theorem volivth 18962
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 730 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  A  e.  dom  vol )
2 mnfxr 10456 . . . . . 6  |-  -oo  e.  RR*
32a1i 10 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  -oo  e.  RR* )
4 iccssxr 10732 . . . . . . 7  |-  ( 0 [,] ( vol `  A
) )  C_  RR*
5 simpr 447 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  ( 0 [,] ( vol `  A
) ) )
64, 5sseldi 3178 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  RR* )
76adantr 451 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR* )
8 iccssxr 10732 . . . . . . . 8  |-  ( 0 [,]  +oo )  C_  RR*
9 volf 18888 . . . . . . . . 9  |-  vol : dom  vol --> ( 0 [,] 
+oo )
109ffvelrni 5664 . . . . . . . 8  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  ( 0 [,] 
+oo ) )
118, 10sseldi 3178 . . . . . . 7  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  RR* )
1211adantr 451 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( vol `  A
)  e.  RR* )
1312adantr 451 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  ( vol `  A
)  e.  RR* )
14 0xr 8878 . . . . . . . . . 10  |-  0  e.  RR*
15 elicc1 10700 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  e.  ( 0 [,] ( vol `  A
) )  <->  ( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A ) ) ) )
1614, 12, 15sylancr 644 . . . . . . . . 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 201 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A
) ) )
1817simp2d 968 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
0  <_  B )
1918adantr 451 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  0  <_  B
)
20 0re 8838 . . . . . . . . 9  |-  0  e.  RR
21 mnflt 10464 . . . . . . . . 9  |-  ( 0  e.  RR  ->  -oo  <  0 )
2220, 21ax-mp 8 . . . . . . . 8  |-  -oo  <  0
23 xrltletr 10488 . . . . . . . 8  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
(  -oo  <  0  /\  0  <_  B )  ->  -oo  <  B ) )
2422, 23mpani 657 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
0  <_  B  ->  -oo 
<  B ) )
252, 14, 24mp3an12 1267 . . . . . 6  |-  ( B  e.  RR*  ->  ( 0  <_  B  ->  -oo  <  B ) )
267, 19, 25sylc 56 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  -oo  <  B )
27 simpr 447 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  <  ( vol `  A ) )
28 xrre2 10499 . . . . 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 1190 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR )
30 volsup2 18960 . . . 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 1182 . . 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 9753 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  RR )
3332ad2antrl 708 . . . . . . . 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 9210 . . . . . . 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 451 . . . . . . 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 10 . . . . . . . 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 9775 . . . . . . . . . 10  |-  ( n  e.  NN  ->  0  <  n )
3837ad2antrl 708 . . . . . . . . 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 9343 . . . . . . . . 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 201 . . . . . . . 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 8977 . . . . . . 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 10731 . . . . . . . 8  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  C_  RR )
4334, 33, 42syl2anc 642 . . . . . . 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 8794 . . . . . . . . 9  |-  RR  C_  CC
45 ssid 3197 . . . . . . . . 9  |-  CC  C_  CC
46 cncfss 18403 . . . . . . . . 9  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR -cn-> RR )  C_  ( RR -cn-> CC ) )
4744, 45, 46mp2an 653 . . . . . . . 8  |-  ( RR
-cn-> RR )  C_  ( RR -cn-> CC )
481adantr 451 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  =  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )
5049volcn 18961 . . . . . . . . 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 642 . . . . . . . 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 3178 . . . . . . 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 3180 . . . . . . . 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 18397 . . . . . . . . . 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 15 . . . . . . . . 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 5663 . . . . . . . . 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 457 . . . . . . . 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 456 . . . . . . 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 5866 . . . . . . . . . . . . . 14  |-  ( y  =  -u n  ->  ( -u n [,] y )  =  ( -u n [,] -u n ) )
6059ineq2d 3370 . . . . . . . . . . . . 13  |-  ( y  =  -u n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] -u n
) ) )
6160fveq2d 5529 . . . . . . . . . . . 12  |-  ( y  =  -u n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] -u n ) ) ) )
62 fvex 5539 . . . . . . . . . . . 12  |-  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) )  e. 
_V
6361, 49, 62fvmpt 5602 . . . . . . . . . . 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 15 . . . . . . . . . 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 3390 . . . . . . . . . . . . . 14  |-  ( A  i^i  ( -u n [,] -u n ) ) 
C_  ( -u n [,] -u n )
6634rexrd 8881 . . . . . . . . . . . . . . 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 10701 . . . . . . . . . . . . . . 15  |-  ( -u n  e.  RR*  ->  ( -u n [,] -u n
)  =  { -u n } )
6866, 67syl 15 . . . . . . . . . . . . . 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 3226 . . . . . . . . . . . . 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 3760 . . . . . . . . . . . . 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 3189 . . . . . . . . . . . 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 18854 . . . . . . . . . . . . . 14  |-  ( -u n  e.  RR  ->  ( vol * `  { -u n } )  =  0 )
7334, 72syl 15 . . . . . . . . . . . . 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 18846 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 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 18893 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 18889 . . . . . . . . . . 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 15 . . . . . . . . . 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 2319 . . . . . . . . 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 451 . . . . . . . . 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 4043 . . . . . . . 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 733 . . . . . . . . . 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 451 . . . . . . . . . . 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 18923 . . . . . . . . . . . . . 14  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  e.  dom  vol )
8634, 33, 85syl2anc 642 . . . . . . . . . . . . 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 18899 . . . . . . . . . . . . 13  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] n )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] n ) )  e. 
dom  vol )
8848, 86, 87syl2anc 642 . . . . . . . . . . . 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 5664 . . . . . . . . . . . . 13  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  ( 0 [,] 
+oo ) )
908, 89sseldi 3178 . . . . . . . . . . . 12  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  RR* )
9188, 90syl 15 . . . . . . . . . . 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 10483 . . . . . . . . . . 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 642 . . . . . . . . . 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 14 . . . . . . . . 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 5866 . . . . . . . . . . . . 13  |-  ( y  =  n  ->  ( -u n [,] y )  =  ( -u n [,] n ) )
9695ineq2d 3370 . . . . . . . . . . . 12  |-  ( y  =  n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] n ) ) )
9796fveq2d 5529 . . . . . . . . . . 11  |-  ( y  =  n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
98 fvex 5539 . . . . . . . . . . 11  |-  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  _V
9997, 49, 98fvmpt 5602 . . . . . . . . . 10  |-  ( n  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
10033, 99syl 15 . . . . . . . . 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 4049 . . . . . . . 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 518 . . . . . . 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 18816 . . . . . 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 3180 . . . . . . . . . 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 5866 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  ( -u n [,] y )  =  ( -u n [,] z ) )
106105ineq2d 3370 . . . . . . . . . . . 12  |-  ( y  =  z  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] z ) ) )
107106fveq2d 5529 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
108 fvex 5539 . . . . . . . . . . 11  |-  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  e.  _V
109107, 49, 108fvmpt 5602 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
110104, 109syl 15 . . . . . . . . 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 2291 . . . . . . . 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 451 . . . . . . . . . . 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 451 . . . . . . . . . . . 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 697 . . . . . . . . . . . 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 18923 . . . . . . . . . . . 12  |-  ( (
-u n  e.  RR  /\  z  e.  RR )  ->  ( -u n [,] z )  e.  dom  vol )
116113, 114, 115syl2anc 642 . . . . . . . . . . 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 18899 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] z )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] z ) )  e. 
dom  vol )
118112, 116, 117syl2anc 642 . . . . . . . . . 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 3389 . . . . . . . . . . 11  |-  ( A  i^i  ( -u n [,] z ) )  C_  A
120119a1i 10 . . . . . . . . . 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 733 . . . . . . . . . 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 3199 . . . . . . . . . . . 12  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
x  C_  A  <->  ( A  i^i  ( -u n [,] z ) )  C_  A ) )
123 fveq2 5525 . . . . . . . . . . . . 13  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  ( vol `  x )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
124123eqeq1d 2291 . . . . . . . . . . . 12  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
( vol `  x
)  =  B  <->  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )
125122, 124anbi12d 691 . . . . . . . . . . 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 2884 . . . . . . . . . 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 1180 . . . . . . . . 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 598 . . . . . . . 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 206 . . . . . . 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 2667 . . . . . 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 14 . . . . 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 598 . . . 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 2667 . . 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 14 . 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 730 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  e.  dom  vol )
136 ssid 3197 . . . 4  |-  A  C_  A
137136a1i 10 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  C_  A
)
138 simpr 447 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  B  =  ( vol `  A ) )
139138eqcomd 2288 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  ( vol `  A )  =  B )
140 sseq1 3199 . . . . 5  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
141 fveq2 5525 . . . . . 6  |-  ( x  =  A  ->  ( vol `  x )  =  ( vol `  A
) )
142141eqeq1d 2291 . . . . 5  |-  ( x  =  A  ->  (
( vol `  x
)  =  B  <->  ( vol `  A )  =  B ) )
143140, 142anbi12d 691 . . . 4  |-  ( x  =  A  ->  (
( x  C_  A  /\  ( vol `  x
)  =  B )  <-> 
( A  C_  A  /\  ( vol `  A
)  =  B ) ) )
144143rspcev 2884 . . 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 1180 . 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 969 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  <_  ( vol `  A
) )
147 xrleloe 10478 . . . 4  |-  ( ( B  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  <_  ( vol `  A
)  <->  ( B  < 
( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
1486, 12, 147syl2anc 642 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <_  ( vol `  A )  <->  ( B  <  ( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
149146, 148mpbid 201 . 2  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <  ( vol `  A )  \/  B  =  ( vol `  A ) ) )
150134, 145, 149mpjaodan 761 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 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   -ucneg 9038   NNcn 9746   [,]cicc 10659   -cn->ccncf 18380   vol *covol 18822   volcvol 18823
This theorem is referenced by:  itg2const2  19096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cmp 17114  df-cncf 18382  df-ovol 18824  df-vol 18825
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