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Theorem uniioombl 21074
Description: A disjoint union of open intervals is measurable. (This proof does not use countable choice, unlike iunmbl 21039.) 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 11392 . . . . 5  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 uniioombl.1 . . . . . 6  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3 inss2 3576 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
4 rexpssxrxp 9433 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
53, 4sstri 3370 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
6 fss 5572 . . . . . 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 5573 . . . . 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 5570 . . . 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 4261 . . 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 3874 . . . . . . . . . . 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 11020 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR+ )
2019rphalfcld 11044 . . . . . . . . . 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 20968 . . . . . . . . 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 1218 . . . . . . . 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 11044 . . . . . . . . 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 7239 . . . . . . . . . 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 21073 . . . . . . . 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 2850 . . . . . . 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 11004 . . . . . . . . . . . . 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 10399 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  2  e.  CC )
43 2ne0 10419 . . . . . . . . . . . . 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 10143 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
( 2  x.  2 ) ) )
46 2t2e4 10476 . . . . . . . . . . . 12  |-  ( 2  x.  2 )  =  4
4746oveq2i 6107 . . . . . . . . . . 11  |-  ( r  /  ( 2  x.  2 ) )  =  ( r  /  4
)
4845, 47syl6eq 2491 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
( r  /  2
)  /  2 )  =  ( r  / 
4 ) )
4948oveq2d 6112 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  ( 4  x.  ( r  /  4
) ) )
50 4cn 10404 . . . . . . . . . . 11  |-  4  e.  CC
5150a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  e.  CC )
52 4ne0 10423 . . . . . . . . . . 11  |-  4  =/=  0
5352a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  4  =/=  0 )
5441, 51, 53divcan2d 10114 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( r  /  4 ) )  =  r )
5549, 54eqtrd 2475 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  /\  r  e.  RR+ )  ->  (
4  x.  ( ( r  /  2 )  /  2 ) )  =  r )
5655oveq2d 6112 . . . . . . 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 4321 . . . . . 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 2804 . . . . 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 3575 . . . . . . . . 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 20974 . . . . . . . 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 1218 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  i^i  U. ran  ( (,)  o.  F
) ) )  e.  RR )
63 difssd 3489 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( z  \  U. ran  ( (,)  o.  F ) )  C_  z )
64 ovolsscl 20974 . . . . . . . 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 1218 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ~P RR  /\  ( vol* `  z )  e.  RR ) )  ->  ( vol* `  ( z  \  U. ran  ( (,)  o.  F
) ) )  e.  RR )
6662, 65readdcld 9418 . . . . . 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 11181 . . . . . 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 2804 . 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 21015 . 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 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721    \ cdif 3330    i^i cin 3332    C_ wss 3333   ~Pcpw 3865   U.cuni 4096  Disj wdisj 4267   class class class wbr 4297    X. cxp 4843   dom cdm 4845   ran crn 4846    o. ccom 4849   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   supcsup 7695   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292   RR*cxr 9422    < clt 9423    <_ cle 9424    - cmin 9600    / cdiv 9998   NNcn 10327   2c2 10376   4c4 10378   RR+crp 10996   (,)cioo 11305    seqcseq 11811   abscabs 12728   vol*covol 20951   volcvol 20952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-disj 4268  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-acn 8117  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-rlim 12972  df-sum 13169  df-rest 14366  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-top 18508  df-bases 18510  df-topon 18511  df-cmp 18995  df-ovol 20953  df-vol 20954
This theorem is referenced by:  uniiccmbl  21075
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