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Theorem uniioombl 21976
Description: A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 21941.) 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 11633 . . . . 5  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 uniioombl.1 . . . . . 6  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3 inss2 3704 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
4 rexpssxrxp 9641 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
53, 4sstri 3498 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
6 fss 5729 . . . . . 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 5731 . . . . 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 5727 . . . 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 4401 . . 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 4006 . . . . . . . . . . 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 757 . . . . . . . . . 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 11255 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR+ )
2019rphalfcld 11279 . . . . . . . . . 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 2443 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
2322ovolgelb 21869 . . . . . . . . 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 1229 . . . . . . . 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 2443 . . . . . . . . 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 11279 . . . . . . . . 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 7442 . . . . . . . . . 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 765 . . . . . . . . 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 766 . . . . . . . . 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 21975 . . . . . . . 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 2939 . . . . . . 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 11239 . . . . . . . . . . . . 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 10615 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  2  e.  CC )
43 2ne0 10635 . . . . . . . . . . . . 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 10358 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
( 2  x.  2 ) ) )
46 2t2e4 10692 . . . . . . . . . . . 12  |-  ( 2  x.  2 )  =  4
4746oveq2i 6292 . . . . . . . . . . 11  |-  ( r  /  ( 2  x.  2 ) )  =  ( r  /  4
)
4845, 47syl6eq 2500 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
4 ) )
4948oveq2d 6297 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  ( 4  x.  ( r  /  4
) ) )
50 4cn 10620 . . . . . . . . . . 11  |-  4  e.  CC
5150a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  e.  CC )
52 4ne0 10639 . . . . . . . . . . 11  |-  4  =/=  0
5352a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  =/=  0 )
5441, 51, 53divcan2d 10329 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( r  /  4 ) )  =  r )
5549, 54eqtrd 2484 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  r )
5655oveq2d 6297 . . . . . . 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 4461 . . . . . 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 2857 . . . . 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 3703 . . . . . . . . 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 21875 . . . . . . . 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 1229 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  e.  RR )
63 difssd 3617 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( z  \  U. ran  ( (,)  o.  F ) )  C_  z )
64 ovolsscl 21875 . . . . . . . 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 1229 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) )  e.  RR )
6662, 65readdcld 9626 . . . . . 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 11416 . . . . . 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 2857 . 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 21916 . 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 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794    \ cdif 3458    i^i cin 3460    C_ wss 3461   ~Pcpw 3997   U.cuni 4234  Disj wdisj 4407   class class class wbr 4437    X. cxp 4987   dom cdm 4989   ran crn 4990    o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281    ^m cmap 7422   supcsup 7902   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   RR*cxr 9630    < clt 9631    <_ cle 9632    - cmin 9810    / cdiv 10213   NNcn 10543   2c2 10592   4c4 10594   RR+crp 11231   (,)cioo 11540    seqcseq 12089   abscabs 13049   vol*covol 21852   volcvol 21853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-n0 10803  df-z 10872  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-rest 14802  df-topgen 14823  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-top 19377  df-bases 19379  df-topon 19380  df-cmp 19865  df-ovol 21854  df-vol 21855
This theorem is referenced by:  uniiccmbl  21977
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