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Theorem uniioombl 20910
Description: A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 20875.) 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 11374 . . . . 5  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 uniioombl.1 . . . . . 6  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3 inss2 3559 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
4 rexpssxrxp 9415 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
53, 4sstri 3353 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
6 fss 5555 . . . . . 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 655 . . . . 5  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
8 fco 5556 . . . . 5  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
91, 7, 8sylancr 656 . . . 4  |-  ( ph  ->  ( (,)  o.  F
) : NN --> ~P RR )
10 frn 5553 . . . 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 4244 . . 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 3857 . . . . . . . . . . 11  |-  ( z  e.  ~P RR  ->  z 
C_  RR )
1514ad2antrl 720 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  z  C_  RR )
1615adantr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  z  C_  RR )
17 simprr 749 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  z )  e.  RR )
1817adantr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  ( vol* `  z )  e.  RR )
19 rphalfcl 11002 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR+ )
2019rphalfcld 11026 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  ( ( r  /  2 )  /  2 )  e.  RR+ )
2120adantl 463 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  e.  RR+ )
22 eqid 2433 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
2322ovolgelb 20804 . . . . . . . . 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 1211 . . . . . . . 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 722 . . . . . . . . 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 722 . . . . . . . . 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 2433 . . . . . . . . 9  |-  U. ran  ( (,)  o.  F )  =  U. ran  ( (,)  o.  F )
3018adantr 462 . . . . . . . . 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 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
r  /  2 )  e.  RR+ )
3231adantr 462 . . . . . . . . . 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 11026 . . . . . . . . 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 7222 . . . . . . . . . 10  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3534ad2antrl 720 . . . . . . . . 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 756 . . . . . . . . 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 757 . . . . . . . . 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 20909 . . . . . . . 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 2835 . . . . . . 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 10986 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e.  CC )
4140adantl 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  r  e.  CC )
42 2cnd 10381 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  2  e.  CC )
43 2ne0 10401 . . . . . . . . . . . . 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 10125 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
( 2  x.  2 ) ) )
46 2t2e4 10458 . . . . . . . . . . . 12  |-  ( 2  x.  2 )  =  4
4746oveq2i 6091 . . . . . . . . . . 11  |-  ( r  /  ( 2  x.  2 ) )  =  ( r  /  4
)
4845, 47syl6eq 2481 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
4 ) )
4948oveq2d 6096 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  ( 4  x.  ( r  /  4
) ) )
50 4cn 10386 . . . . . . . . . . 11  |-  4  e.  CC
5150a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  e.  CC )
52 4ne0 10405 . . . . . . . . . . 11  |-  4  =/=  0
5352a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  =/=  0 )
5441, 51, 53divcan2d 10096 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( r  /  4 ) )  =  r )
5549, 54eqtrd 2465 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  r )
5655oveq2d 6096 . . . . . . 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 4304 . . . . . 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 2789 . . . . 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 3558 . . . . . . . . 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 20810 . . . . . . . 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 1211 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  e.  RR )
63 difssd 3472 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( z  \  U. ran  ( (,)  o.  F ) )  C_  z )
64 ovolsscl 20810 . . . . . . . 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 1211 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) )  e.  RR )
6662, 65readdcld 9400 . . . . . 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 11163 . . . . . 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 654 . . . . 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 610 . . 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 2789 . 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 20851 . 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 657 1  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  e. 
dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705   E.wrex 2706    \ cdif 3313    i^i cin 3315    C_ wss 3316   ~Pcpw 3848   U.cuni 4079  Disj wdisj 4250   class class class wbr 4280    X. cxp 4825   dom cdm 4827   ran crn 4828    o. ccom 4831   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   supcsup 7678   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    + caddc 9272    x. cmul 9274   RR*cxr 9404    < clt 9405    <_ cle 9406    - cmin 9582    / cdiv 9980   NNcn 10309   2c2 10358   4c4 10360   RR+crp 10978   (,)cioo 11287    seqcseq 11789   abscabs 12706   vol*covol 20787   volcvol 20788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-acn 8100  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-n0 10567  df-z 10634  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-fl 11625  df-seq 11790  df-exp 11849  df-hash 12087  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-clim 12949  df-rlim 12950  df-sum 13147  df-rest 14343  df-topgen 14364  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-top 18344  df-bases 18346  df-topon 18347  df-cmp 18831  df-ovol 20789  df-vol 20790
This theorem is referenced by:  uniiccmbl  20911
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