MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniioombl Structured version   Visualization version   Unicode version

Theorem uniioombl 22547
Description: A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 22506.) Lemma 565Ca of [Fremlin5] p. 214. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypotheses
Ref Expression
uniioombl.1  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
uniioombl.2  |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )
uniioombl.3  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
Assertion
Ref Expression
uniioombl  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  e. 
dom  vol )
Distinct variable groups:    x, F    ph, x
Allowed substitution hint:    S( x)

Proof of Theorem uniioombl
Dummy variables  f 
r  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioof 11732 . . . . 5  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 uniioombl.1 . . . . . 6  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3 inss2 3653 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
4 rexpssxrxp 9685 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
53, 4sstri 3441 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
6 fss 5737 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
72, 5, 6sylancl 668 . . . . 5  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
8 fco 5739 . . . . 5  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
91, 7, 8sylancr 669 . . . 4  |-  ( ph  ->  ( (,)  o.  F
) : NN --> ~P RR )
10 frn 5735 . . . 4  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  ran  ( (,)  o.  F )  C_  ~P RR )
119, 10syl 17 . . 3  |-  ( ph  ->  ran  ( (,)  o.  F )  C_  ~P RR )
12 sspwuni 4367 . . 3  |-  ( ran  ( (,)  o.  F
)  C_  ~P RR  <->  U.
ran  ( (,)  o.  F )  C_  RR )
1311, 12sylib 200 . 2  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  RR )
14 elpwi 3960 . . . . . . . . . . 11  |-  ( z  e.  ~P RR  ->  z 
C_  RR )
1514ad2antrl 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  z  C_  RR )
1615adantr 467 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  z  C_  RR )
17 simprr 766 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  z )  e.  RR )
1817adantr 467 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  ( vol* `  z )  e.  RR )
19 rphalfcl 11327 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR+ )
2019rphalfcld 11353 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  ( ( r  /  2 )  /  2 )  e.  RR+ )
2120adantl 468 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  e.  RR+ )
22 eqid 2451 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
2322ovolgelb 22433 . . . . . . . . 9  |-  ( ( z  C_  RR  /\  ( vol* `  z )  e.  RR  /\  (
( r  /  2
)  /  2 )  e.  RR+ )  ->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( z  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  z )  +  ( ( r  /  2
)  /  2 ) ) ) )
2416, 18, 21, 23syl3anc 1268 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( z  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  z )  +  ( ( r  /  2
)  /  2 ) ) ) )
252ad3antrrr 736 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
26 uniioombl.2 . . . . . . . . . 10  |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )
2726ad3antrrr 736 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  -> Disj  x  e.  NN  ( (,) `  ( F `  x )
) )
28 uniioombl.3 . . . . . . . . 9  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
29 eqid 2451 . . . . . . . . 9  |-  U. ran  ( (,)  o.  F )  =  U. ran  ( (,)  o.  F )
3018adantr 467 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  ->  ( vol* `  z )  e.  RR )
3119adantl 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
r  /  2 )  e.  RR+ )
3231adantr 467 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  ->  (
r  /  2 )  e.  RR+ )
3332rphalfcld 11353 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  ->  (
( r  /  2
)  /  2 )  e.  RR+ )
34 elmapi 7493 . . . . . . . . . 10  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3534ad2antrl 734 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
36 simprrl 774 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  ->  z  C_ 
U. ran  ( (,)  o.  f ) )
37 simprrr 775 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  ->  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  z )  +  ( ( r  /  2
)  /  2 ) ) )
3825, 27, 28, 29, 30, 33, 35, 36, 22, 37uniioombllem6 22546 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  /\  (
f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  (
z  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  )  <_  ( ( vol* `  z )  +  ( ( r  /  2 )  / 
2 ) ) ) ) )  ->  (
( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( ( vol* `  z )  +  ( 4  x.  ( ( r  / 
2 )  /  2
) ) ) )
3924, 38rexlimddv 2883 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( ( vol* `  z )  +  ( 4  x.  ( ( r  / 
2 )  /  2
) ) ) )
40 rpcn 11310 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e.  CC )
4140adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  r  e.  CC )
42 2cnd 10682 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  2  e.  CC )
43 2ne0 10702 . . . . . . . . . . . . 13  |-  2  =/=  0
4443a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  2  =/=  0 )
4541, 42, 42, 44, 44divdiv1d 10414 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
( 2  x.  2 ) ) )
46 2t2e4 10759 . . . . . . . . . . . 12  |-  ( 2  x.  2 )  =  4
4746oveq2i 6301 . . . . . . . . . . 11  |-  ( r  /  ( 2  x.  2 ) )  =  ( r  /  4
)
4845, 47syl6eq 2501 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
4 ) )
4948oveq2d 6306 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  ( 4  x.  ( r  /  4
) ) )
50 4cn 10687 . . . . . . . . . . 11  |-  4  e.  CC
5150a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  e.  CC )
52 4ne0 10706 . . . . . . . . . . 11  |-  4  =/=  0
5352a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  =/=  0 )
5441, 51, 53divcan2d 10385 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( r  /  4 ) )  =  r )
5549, 54eqtrd 2485 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  r )
5655oveq2d 6306 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( vol* `  z )  +  ( 4  x.  ( ( r  /  2 )  /  2 ) ) )  =  ( ( vol* `  z
)  +  r ) )
5739, 56breqtrd 4427 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( ( vol* `  z )  +  r ) )
5857ralrimiva 2802 . . . . 5  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  A. r  e.  RR+  ( ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( ( vol* `  z )  +  r ) )
59 inss1 3652 . . . . . . . . 9  |-  ( z  i^i  U. ran  ( (,)  o.  F ) ) 
C_  z
6059a1i 11 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( z  i^i  U. ran  ( (,)  o.  F ) )  C_  z )
61 ovolsscl 22439 . . . . . . . 8  |-  ( ( ( z  i^i  U. ran  ( (,)  o.  F
) )  C_  z  /\  z  C_  RR  /\  ( vol* `  z
)  e.  RR )  ->  ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  e.  RR )
6260, 15, 17, 61syl3anc 1268 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  e.  RR )
63 difssd 3561 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( z  \  U. ran  ( (,)  o.  F ) )  C_  z )
64 ovolsscl 22439 . . . . . . . 8  |-  ( ( ( z  \  U. ran  ( (,)  o.  F
) )  C_  z  /\  z  C_  RR  /\  ( vol* `  z
)  e.  RR )  ->  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) )  e.  RR )
6563, 15, 17, 64syl3anc 1268 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) )  e.  RR )
6662, 65readdcld 9670 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( ( vol* `  ( z  i^i  U. ran  ( (,) 
o.  F ) ) )  +  ( vol* `  ( z  \  U. ran  ( (,) 
o.  F ) ) ) )  e.  RR )
67 alrple 11499 . . . . . 6  |-  ( ( ( ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  e.  RR  /\  ( vol* `  z )  e.  RR )  -> 
( ( ( vol* `  ( z  i^i  U. ran  ( (,) 
o.  F ) ) )  +  ( vol* `  ( z  \  U. ran  ( (,) 
o.  F ) ) ) )  <_  ( vol* `  z )  <->  A. r  e.  RR+  (
( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( ( vol* `  z )  +  r ) ) )
6866, 17, 67syl2anc 667 . . . . 5  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( ( ( vol* `  (
z  i^i  U. ran  ( (,)  o.  F ) ) )  +  ( vol* `  ( z  \  U. ran  ( (,) 
o.  F ) ) ) )  <_  ( vol* `  z )  <->  A. r  e.  RR+  (
( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( ( vol* `  z )  +  r ) ) )
6958, 68mpbird 236 . . . 4  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( ( vol* `  ( z  i^i  U. ran  ( (,) 
o.  F ) ) )  +  ( vol* `  ( z  \  U. ran  ( (,) 
o.  F ) ) ) )  <_  ( vol* `  z ) )
7069expr 620 . . 3  |-  ( (
ph  /\  z  e.  ~P RR )  ->  (
( vol* `  z )  e.  RR  ->  ( ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( vol* `  z ) ) )
7170ralrimiva 2802 . 2  |-  ( ph  ->  A. z  e.  ~P  RR ( ( vol* `  z )  e.  RR  ->  ( ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( vol* `  z ) ) )
72 ismbl2 22481 . 2  |-  ( U. ran  ( (,)  o.  F
)  e.  dom  vol  <->  ( U. ran  ( (,)  o.  F )  C_  RR  /\ 
A. z  e.  ~P  RR ( ( vol* `  z )  e.  RR  ->  ( ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  +  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) ) )  <_  ( vol* `  z ) ) ) )
7313, 71, 72sylanbrc 670 1  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  e. 
dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    \ cdif 3401    i^i cin 3403    C_ wss 3404   ~Pcpw 3951   U.cuni 4198  Disj wdisj 4373   class class class wbr 4402    X. cxp 4832   dom cdm 4834   ran crn 4835    o. ccom 4838   -->wf 5578   ` cfv 5582  (class class class)co 6290    ^m cmap 7472   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   2c2 10659   4c4 10661   RR+crp 11302   (,)cioo 11635    seqcseq 12213   abscabs 13297   vol*covol 22413   volcvol 22415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-rest 15321  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-top 19921  df-bases 19922  df-topon 19923  df-cmp 20402  df-ovol 22416  df-vol 22418
This theorem is referenced by:  uniiccmbl  22548
  Copyright terms: Public domain W3C validator