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Theorem volivth 19499
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 731 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  A  e.  dom  vol )
2 mnfxr 10714 . . . . . 6  |-  -oo  e.  RR*
32a1i 11 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  -oo  e.  RR* )
4 iccssxr 10993 . . . . . . 7  |-  ( 0 [,] ( vol `  A
) )  C_  RR*
5 simpr 448 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  ( 0 [,] ( vol `  A
) ) )
64, 5sseldi 3346 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  e.  RR* )
76adantr 452 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  e.  RR* )
8 iccssxr 10993 . . . . . . . 8  |-  ( 0 [,]  +oo )  C_  RR*
9 volf 19425 . . . . . . . . 9  |-  vol : dom  vol --> ( 0 [,] 
+oo )
109ffvelrni 5869 . . . . . . . 8  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  ( 0 [,] 
+oo ) )
118, 10sseldi 3346 . . . . . . 7  |-  ( A  e.  dom  vol  ->  ( vol `  A )  e.  RR* )
1211adantr 452 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( vol `  A
)  e.  RR* )
1312adantr 452 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  ( vol `  A
)  e.  RR* )
14 0xr 9131 . . . . . . . . . 10  |-  0  e.  RR*
15 elicc1 10960 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  e.  ( 0 [,] ( vol `  A
) )  <->  ( B  e.  RR*  /\  0  <_  B  /\  B  <_  ( vol `  A ) ) ) )
1614, 12, 15sylancr 645 . . . . . . . . 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 202 . . . . . . . 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 452 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  0  <_  B
)
20 0re 9091 . . . . . . . . 9  |-  0  e.  RR
21 mnflt 10722 . . . . . . . . 9  |-  ( 0  e.  RR  ->  -oo  <  0 )
2220, 21ax-mp 8 . . . . . . . 8  |-  -oo  <  0
23 xrltletr 10747 . . . . . . . 8  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR*  /\  B  e. 
RR* )  ->  (
(  -oo  <  0  /\  0  <_  B )  ->  -oo  <  B ) )
2422, 23mpani 658 . . . . . . 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 448 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  B  <  ( vol `  A ) )
28 xrre2 10758 . . . . 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 19497 . . . 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 10007 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  RR )
3332ad2antrl 709 . . . . . 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 )
3433renegcld 9464 . . . . 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 )
3529adantr 452 . . . . 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 )
3620a1i 11 . . . . . 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 )
37 nngt0 10029 . . . . . . . 8  |-  ( n  e.  NN  ->  0  <  n )
3837ad2antrl 709 . . . . . . 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 )
3933lt0neg2d 9597 . . . . . . 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 ) )
4038, 39mpbid 202 . . . . . 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 )
4134, 36, 33, 40, 38lttrd 9231 . . . . 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 )
42 iccssre 10992 . . . . . 6  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  C_  RR )
4334, 33, 42syl2anc 643 . . . . 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 )
44 ax-resscn 9047 . . . . . . 7  |-  RR  C_  CC
45 ssid 3367 . . . . . . 7  |-  CC  C_  CC
46 cncfss 18929 . . . . . . 7  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR -cn-> RR )  C_  ( RR -cn-> CC ) )
4744, 45, 46mp2an 654 . . . . . 6  |-  ( RR
-cn-> RR )  C_  ( RR -cn-> CC )
481adantr 452 . . . . . . 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 )
49 eqid 2436 . . . . . . . 8  |-  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )  =  ( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) )
5049volcn 19498 . . . . . . 7  |-  ( ( 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 643 . . . . . 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 ) )
5247, 51sseldi 3346 . . . . 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 ) )
5343sselda 3348 . . . . . 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 )
54 cncff 18923 . . . . . . . 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 )
5551, 54syl 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 )
5655ffvelrnda 5870 . . . . . 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 )
5753, 56syldan 457 . . . . 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 )
58 oveq2 6089 . . . . . . . . . . . 12  |-  ( y  =  -u n  ->  ( -u n [,] y )  =  ( -u n [,] -u n ) )
5958ineq2d 3542 . . . . . . . . . . 11  |-  ( y  =  -u n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] -u n
) ) )
6059fveq2d 5732 . . . . . . . . . 10  |-  ( y  =  -u n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] -u n ) ) ) )
61 fvex 5742 . . . . . . . . . 10  |-  ( vol `  ( A  i^i  ( -u n [,] -u n
) ) )  e. 
_V
6260, 49, 61fvmpt 5806 . . . . . . . . 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
) ) ) )
6334, 62syl 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
) ) ) )
64 inss2 3562 . . . . . . . . . . . 12  |-  ( A  i^i  ( -u n [,] -u n ) ) 
C_  ( -u n [,] -u n )
6534rexrd 9134 . . . . . . . . . . . . 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* )
66 iccid 10961 . . . . . . . . . . . . 13  |-  ( -u n  e.  RR*  ->  ( -u n [,] -u n
)  =  { -u n } )
6765, 66syl 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 } )
6864, 67syl5sseq 3396 . . . . . . . . . . 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 } )
6934snssd 3943 . . . . . . . . . . 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 )
7068, 69sstrd 3358 . . . . . . . . . 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 )
71 ovolsn 19391 . . . . . . . . . . . 12  |-  ( -u n  e.  RR  ->  ( vol * `  { -u n } )  =  0 )
7234, 71syl 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 )
73 ovolssnul 19383 . . . . . . . . . . 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 )
7468, 69, 72, 73syl3anc 1184 . . . . . . . . . 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 )
75 nulmbl 19430 . . . . . . . . . 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 )
7670, 74, 75syl2anc 643 . . . . . . . . 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 )
77 mblvol 19426 . . . . . . . . 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 ) ) ) )
7876, 77syl 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
) ) ) )
7963, 78, 743eqtrd 2472 . . . . . . 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 )
8019adantr 452 . . . . . . 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 )
8179, 80eqbrtrd 4232 . . . . . 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 )
82 simprr 734 . . . . . . . 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 ) ) ) )
837adantr 452 . . . . . . . . 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* )
84 iccmbl 19460 . . . . . . . . . . . 12  |-  ( (
-u n  e.  RR  /\  n  e.  RR )  ->  ( -u n [,] n )  e.  dom  vol )
8534, 33, 84syl2anc 643 . . . . . . . . . . 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 )
86 inmbl 19436 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] n )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] n ) )  e. 
dom  vol )
8748, 85, 86syl2anc 643 . . . . . . . . . 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 )
889ffvelrni 5869 . . . . . . . . . . 11  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  ( 0 [,] 
+oo ) )
898, 88sseldi 3346 . . . . . . . . . 10  |-  ( ( A  i^i  ( -u n [,] n ) )  e.  dom  vol  ->  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  RR* )
9087, 89syl 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* )
91 xrltle 10742 . . . . . . . . 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
) ) ) ) )
9283, 90, 91syl2anc 643 . . . . . . . 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
) ) ) ) )
9382, 92mpd 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 ) ) ) )
94 oveq2 6089 . . . . . . . . . . 11  |-  ( y  =  n  ->  ( -u n [,] y )  =  ( -u n [,] n ) )
9594ineq2d 3542 . . . . . . . . . 10  |-  ( y  =  n  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] n ) ) )
9695fveq2d 5732 . . . . . . . . 9  |-  ( y  =  n  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
97 fvex 5742 . . . . . . . . 9  |-  ( vol `  ( A  i^i  ( -u n [,] n ) ) )  e.  _V
9896, 49, 97fvmpt 5806 . . . . . . . 8  |-  ( n  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  n )  =  ( vol `  ( A  i^i  ( -u n [,] n ) ) ) )
9933, 98syl 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 ) ) ) )
10093, 99breqtrrd 4238 . . . . . 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 ) )
10181, 100jca 519 . . . . 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 ) ) )
10234, 33, 35, 41, 43, 52, 57, 101ivthle 19353 . . . 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 )
10343sselda 3348 . . . . . . . 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 )
104 oveq2 6089 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( -u n [,] y )  =  ( -u n [,] z ) )
105104ineq2d 3542 . . . . . . . . . 10  |-  ( y  =  z  ->  ( A  i^i  ( -u n [,] y ) )  =  ( A  i^i  ( -u n [,] z ) ) )
106105fveq2d 5732 . . . . . . . . 9  |-  ( y  =  z  ->  ( vol `  ( A  i^i  ( -u n [,] y
) ) )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
107 fvex 5742 . . . . . . . . 9  |-  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  e.  _V
108106, 49, 107fvmpt 5806 . . . . . . . 8  |-  ( z  e.  RR  ->  (
( y  e.  RR  |->  ( vol `  ( A  i^i  ( -u n [,] y ) ) ) ) `  z )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
109103, 108syl 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 ) ) ) )
110109eqeq1d 2444 . . . . . 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 ) )
11148adantr 452 . . . . . . . . 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 )
11234adantr 452 . . . . . . . . . 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 )
113103adantrr 698 . . . . . . . . . 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 )
114 iccmbl 19460 . . . . . . . . . 10  |-  ( (
-u n  e.  RR  /\  z  e.  RR )  ->  ( -u n [,] z )  e.  dom  vol )
115112, 113, 114syl2anc 643 . . . . . . . . 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 )
116 inmbl 19436 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  ( -u n [,] z )  e.  dom  vol )  ->  ( A  i^i  ( -u n [,] z ) )  e. 
dom  vol )
117111, 115, 116syl2anc 643 . . . . . . . 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 )
118 inss1 3561 . . . . . . . . 9  |-  ( A  i^i  ( -u n [,] z ) )  C_  A
119118a1i 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
)
120 simprr 734 . . . . . . . 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 )
121 sseq1 3369 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
x  C_  A  <->  ( A  i^i  ( -u n [,] z ) )  C_  A ) )
122 fveq2 5728 . . . . . . . . . . 11  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  ( vol `  x )  =  ( vol `  ( A  i^i  ( -u n [,] z ) ) ) )
123122eqeq1d 2444 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( -u n [,] z
) )  ->  (
( vol `  x
)  =  B  <->  ( vol `  ( A  i^i  ( -u n [,] z ) ) )  =  B ) )
124121, 123anbi12d 692 . . . . . . . . 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 ) ) )
125124rspcev 3052 . . . . . . . 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 ) )
126117, 119, 120, 125syl12anc 1182 . . . . . . 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 ) )
127126expr 599 . . . . . 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 ) ) )
128110, 127sylbid 207 . . . . 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 ) ) )
129128rexlimdva 2830 . . . 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 ) ) )
130102, 129mpd 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 ) )
13131, 130rexlimddv 2834 . 2  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  <  ( vol `  A ) )  ->  E. x  e.  dom  vol ( x  C_  A  /\  ( vol `  x
)  =  B ) )
132 simpll 731 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  e.  dom  vol )
133 ssid 3367 . . . 4  |-  A  C_  A
134133a1i 11 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  A  C_  A
)
135 simpr 448 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  B  =  ( vol `  A ) )
136135eqcomd 2441 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  ( 0 [,] ( vol `  A ) ) )  /\  B  =  ( vol `  A ) )  ->  ( vol `  A )  =  B )
137 sseq1 3369 . . . . 5  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
138 fveq2 5728 . . . . . 6  |-  ( x  =  A  ->  ( vol `  x )  =  ( vol `  A
) )
139138eqeq1d 2444 . . . . 5  |-  ( x  =  A  ->  (
( vol `  x
)  =  B  <->  ( vol `  A )  =  B ) )
140137, 139anbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( x  C_  A  /\  ( vol `  x
)  =  B )  <-> 
( A  C_  A  /\  ( vol `  A
)  =  B ) ) )
141140rspcev 3052 . . 3  |-  ( ( A  e.  dom  vol  /\  ( A  C_  A  /\  ( vol `  A
)  =  B ) )  ->  E. x  e.  dom  vol ( x 
C_  A  /\  ( vol `  x )  =  B ) )
142132, 134, 136, 141syl12anc 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 ) )
14317simp3d 971 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  ->  B  <_  ( vol `  A
) )
144 xrleloe 10737 . . . 4  |-  ( ( B  e.  RR*  /\  ( vol `  A )  e. 
RR* )  ->  ( B  <_  ( vol `  A
)  <->  ( B  < 
( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
1456, 12, 144syl2anc 643 . . 3  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <_  ( vol `  A )  <->  ( B  <  ( vol `  A
)  \/  B  =  ( vol `  A
) ) ) )
146143, 145mpbid 202 . 2  |-  ( ( A  e.  dom  vol  /\  B  e.  ( 0 [,] ( vol `  A
) ) )  -> 
( B  <  ( vol `  A )  \/  B  =  ( vol `  A ) ) )
147131, 142, 146mpjaodan 762 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319    C_ wss 3320   {csn 3814   class class class wbr 4212    e. cmpt 4266   dom cdm 4878   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990    +oocpnf 9117    -oocmnf 9118   RR*cxr 9119    < clt 9120    <_ cle 9121   -ucneg 9292   NNcn 10000   [,]cicc 10919   -cn->ccncf 18906   vol *covol 19359   volcvol 19360
This theorem is referenced by:  itg2const2  19633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cc 8315  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-rest 13650  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-cmp 17450  df-cncf 18908  df-ovol 19361  df-vol 19362
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