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Theorem uniioombl 21866
Description: A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 21831.) 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 11634 . . . . 5  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 uniioombl.1 . . . . . 6  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3 inss2 3724 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
4 rexpssxrxp 9650 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
53, 4sstri 3518 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
6 fss 5745 . . . . . 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 662 . . . . 5  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
8 fco 5747 . . . . 5  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
91, 7, 8sylancr 663 . . . 4  |-  ( ph  ->  ( (,)  o.  F
) : NN --> ~P RR )
10 frn 5743 . . . 4  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  ran  ( (,)  o.  F )  C_  ~P RR )
119, 10syl 16 . . 3  |-  ( ph  ->  ran  ( (,)  o.  F )  C_  ~P RR )
12 sspwuni 4417 . . 3  |-  ( ran  ( (,)  o.  F
)  C_  ~P RR  <->  U.
ran  ( (,)  o.  F )  C_  RR )
1311, 12sylib 196 . 2  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  RR )
14 elpwi 4025 . . . . . . . . . . 11  |-  ( z  e.  ~P RR  ->  z 
C_  RR )
1514ad2antrl 727 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  z  C_  RR )
1615adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  z  C_  RR )
17 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  z )  e.  RR )
1817adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  ( vol* `  z )  e.  RR )
19 rphalfcl 11256 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR+ )
2019rphalfcld 11280 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  ( ( r  /  2 )  /  2 )  e.  RR+ )
2120adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  e.  RR+ )
22 eqid 2467 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
2322ovolgelb 21759 . . . . . . . . 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 1228 . . . . . . . 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 729 . . . . . . . . 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 729 . . . . . . . . 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 2467 . . . . . . . . 9  |-  U. ran  ( (,)  o.  F )  =  U. ran  ( (,)  o.  F )
3018adantr 465 . . . . . . . . 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 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
r  /  2 )  e.  RR+ )
3231adantr 465 . . . . . . . . . 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 11280 . . . . . . . . 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 7452 . . . . . . . . . 10  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3534ad2antrl 727 . . . . . . . . 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 763 . . . . . . . . 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 764 . . . . . . . . 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 21865 . . . . . . . 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 2963 . . . . . . 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 11240 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e.  CC )
4140adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  r  e.  CC )
42 2cnd 10620 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  2  e.  CC )
43 2ne0 10640 . . . . . . . . . . . . 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 10363 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
( 2  x.  2 ) ) )
46 2t2e4 10697 . . . . . . . . . . . 12  |-  ( 2  x.  2 )  =  4
4746oveq2i 6306 . . . . . . . . . . 11  |-  ( r  /  ( 2  x.  2 ) )  =  ( r  /  4
)
4845, 47syl6eq 2524 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
4 ) )
4948oveq2d 6311 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  ( 4  x.  ( r  /  4
) ) )
50 4cn 10625 . . . . . . . . . . 11  |-  4  e.  CC
5150a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  e.  CC )
52 4ne0 10644 . . . . . . . . . . 11  |-  4  =/=  0
5352a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  =/=  0 )
5441, 51, 53divcan2d 10334 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( r  /  4 ) )  =  r )
5549, 54eqtrd 2508 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  r )
5655oveq2d 6311 . . . . . . 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 4477 . . . . . 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 2881 . . . . 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 3723 . . . . . . . . 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 21765 . . . . . . . 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 1228 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  e.  RR )
63 difssd 3637 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( z  \  U. ran  ( (,)  o.  F ) )  C_  z )
64 ovolsscl 21765 . . . . . . . 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 1228 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) )  e.  RR )
6662, 65readdcld 9635 . . . . . 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 11417 . . . . . 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 661 . . . . 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 232 . . . 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 615 . . 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 2881 . 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 21806 . 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 664 1  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  e. 
dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    \ cdif 3478    i^i cin 3480    C_ wss 3481   ~Pcpw 4016   U.cuni 4251  Disj wdisj 4423   class class class wbr 4453    X. cxp 5003   dom cdm 5005   ran crn 5006    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   supcsup 7912   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   RR*cxr 9639    < clt 9640    <_ cle 9641    - cmin 9817    / cdiv 10218   NNcn 10548   2c2 10597   4c4 10599   RR+crp 11232   (,)cioo 11541    seqcseq 12087   abscabs 13047   vol*covol 21742   volcvol 21743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-rest 14695  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-top 19268  df-bases 19270  df-topon 19271  df-cmp 19755  df-ovol 21744  df-vol 21745
This theorem is referenced by:  uniiccmbl  21867
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