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Theorem uniioombllem5 21869
Description: Lemma for uniioombl 21871. (Contributed by Mario Carneiro, 25-Aug-2014.)
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 ) )
uniioombl.a  |-  A  = 
U. ran  ( (,)  o.  F )
uniioombl.e  |-  ( ph  ->  ( vol* `  E )  e.  RR )
uniioombl.c  |-  ( ph  ->  C  e.  RR+ )
uniioombl.g  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
uniioombl.s  |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )
uniioombl.t  |-  T  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
uniioombl.v  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C
) )
uniioombl.m  |-  ( ph  ->  M  e.  NN )
uniioombl.m2  |-  ( ph  ->  ( abs `  (
( T `  M
)  -  sup ( ran  T ,  RR* ,  <  ) ) )  <  C
)
uniioombl.k  |-  K  = 
U. ( ( (,) 
o.  G ) "
( 1 ... M
) )
uniioombl.n  |-  ( ph  ->  N  e.  NN )
uniioombl.n2  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( abs `  ( sum_ i  e.  ( 1 ... N ) ( vol* `  (
( (,) `  ( F `  i )
)  i^i  ( (,) `  ( G `  j
) ) ) )  -  ( vol* `  ( ( (,) `  ( G `  j )
)  i^i  A )
) ) )  < 
( C  /  M
) )
uniioombl.l  |-  L  = 
U. ( ( (,) 
o.  F ) "
( 1 ... N
) )
Assertion
Ref Expression
uniioombllem5  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <_  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
Distinct variable groups:    i, j, x, F    i, G, j, x    j, K, x    A, j, x    C, i, j, x    i, M, j, x    i, N, j    ph, i, j, x    T, i, j, x
Allowed substitution hints:    A( i)    S( x, i, j)    E( x, i, j)    K( i)    L( x, i, j)    N( x)

Proof of Theorem uniioombllem5
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 inss1 3703 . . . . 5  |-  ( E  i^i  A )  C_  E
21a1i 11 . . . 4  |-  ( ph  ->  ( E  i^i  A
)  C_  E )
3 uniioombl.s . . . . 5  |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )
4 uniioombl.g . . . . . . . 8  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
54uniiccdif 21860 . . . . . . 7  |-  ( ph  ->  ( U. ran  ( (,)  o.  G )  C_  U.
ran  ( [,]  o.  G )  /\  ( vol* `  ( U. ran  ( [,]  o.  G
)  \  U. ran  ( (,)  o.  G ) ) )  =  0 ) )
65simpld 459 . . . . . 6  |-  ( ph  ->  U. ran  ( (,) 
o.  G )  C_  U.
ran  ( [,]  o.  G ) )
7 ovolficcss 21754 . . . . . . 7  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  G ) 
C_  RR )
84, 7syl 16 . . . . . 6  |-  ( ph  ->  U. ran  ( [,] 
o.  G )  C_  RR )
96, 8sstrd 3499 . . . . 5  |-  ( ph  ->  U. ran  ( (,) 
o.  G )  C_  RR )
103, 9sstrd 3499 . . . 4  |-  ( ph  ->  E  C_  RR )
11 uniioombl.e . . . 4  |-  ( ph  ->  ( vol* `  E )  e.  RR )
12 ovolsscl 21770 . . . 4  |-  ( ( ( E  i^i  A
)  C_  E  /\  E  C_  RR  /\  ( vol* `  E )  e.  RR )  -> 
( vol* `  ( E  i^i  A ) )  e.  RR )
132, 10, 11, 12syl3anc 1229 . . 3  |-  ( ph  ->  ( vol* `  ( E  i^i  A ) )  e.  RR )
14 difssd 3617 . . . 4  |-  ( ph  ->  ( E  \  A
)  C_  E )
15 ovolsscl 21770 . . . 4  |-  ( ( ( E  \  A
)  C_  E  /\  E  C_  RR  /\  ( vol* `  E )  e.  RR )  -> 
( vol* `  ( E  \  A ) )  e.  RR )
1614, 10, 11, 15syl3anc 1229 . . 3  |-  ( ph  ->  ( vol* `  ( E  \  A ) )  e.  RR )
1713, 16readdcld 9626 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  e.  RR )
18 inss1 3703 . . . . . 6  |-  ( K  i^i  A )  C_  K
1918a1i 11 . . . . 5  |-  ( ph  ->  ( K  i^i  A
)  C_  K )
20 uniioombl.k . . . . . . . 8  |-  K  = 
U. ( ( (,) 
o.  G ) "
( 1 ... M
) )
21 imassrn 5338 . . . . . . . . 9  |-  ( ( (,)  o.  G )
" ( 1 ... M ) )  C_  ran  ( (,)  o.  G
)
2221unissi 4257 . . . . . . . 8  |-  U. (
( (,)  o.  G
) " ( 1 ... M ) ) 
C_  U. ran  ( (,) 
o.  G )
2320, 22eqsstri 3519 . . . . . . 7  |-  K  C_  U.
ran  ( (,)  o.  G )
2423a1i 11 . . . . . 6  |-  ( ph  ->  K  C_  U. ran  ( (,)  o.  G ) )
2524, 9sstrd 3499 . . . . 5  |-  ( ph  ->  K  C_  RR )
26 uniioombl.1 . . . . . . . 8  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
27 uniioombl.2 . . . . . . . 8  |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )
28 uniioombl.3 . . . . . . . 8  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
29 uniioombl.a . . . . . . . 8  |-  A  = 
U. ran  ( (,)  o.  F )
30 uniioombl.c . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
31 uniioombl.t . . . . . . . 8  |-  T  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
32 uniioombl.v . . . . . . . 8  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C
) )
3326, 27, 28, 29, 11, 30, 4, 3, 31, 32uniioombllem1 21863 . . . . . . 7  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
34 ssid 3508 . . . . . . . 8  |-  U. ran  ( (,)  o.  G ) 
C_  U. ran  ( (,) 
o.  G )
3531ovollb 21763 . . . . . . . 8  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  U. ran  ( (,)  o.  G
)  C_  U. ran  ( (,)  o.  G ) )  ->  ( vol* `  U. ran  ( (,) 
o.  G ) )  <_  sup ( ran  T ,  RR* ,  <  )
)
364, 34, 35sylancl 662 . . . . . . 7  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  <_  sup ( ran  T ,  RR* ,  <  ) )
37 ovollecl 21767 . . . . . . 7  |-  ( ( U. ran  ( (,) 
o.  G )  C_  RR  /\  sup ( ran 
T ,  RR* ,  <  )  e.  RR  /\  ( vol* `  U. ran  ( (,)  o.  G ) )  <_  sup ( ran  T ,  RR* ,  <  ) )  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  e.  RR )
389, 33, 36, 37syl3anc 1229 . . . . . 6  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  e.  RR )
39 ovolsscl 21770 . . . . . 6  |-  ( ( K  C_  U. ran  ( (,)  o.  G )  /\  U.
ran  ( (,)  o.  G )  C_  RR  /\  ( vol* `  U. ran  ( (,)  o.  G ) )  e.  RR )  ->  ( vol* `  K )  e.  RR )
4024, 9, 38, 39syl3anc 1229 . . . . 5  |-  ( ph  ->  ( vol* `  K )  e.  RR )
41 ovolsscl 21770 . . . . 5  |-  ( ( ( K  i^i  A
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  i^i  A ) )  e.  RR )
4219, 25, 40, 41syl3anc 1229 . . . 4  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  e.  RR )
43 difssd 3617 . . . . 5  |-  ( ph  ->  ( K  \  A
)  C_  K )
44 ovolsscl 21770 . . . . 5  |-  ( ( ( K  \  A
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  A ) )  e.  RR )
4543, 25, 40, 44syl3anc 1229 . . . 4  |-  ( ph  ->  ( vol* `  ( K  \  A ) )  e.  RR )
4642, 45readdcld 9626 . . 3  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  e.  RR )
4730rpred 11265 . . . 4  |-  ( ph  ->  C  e.  RR )
4847, 47readdcld 9626 . . 3  |-  ( ph  ->  ( C  +  C
)  e.  RR )
4946, 48readdcld 9626 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  e.  RR )
50 4re 10618 . . . 4  |-  4  e.  RR
51 remulcl 9580 . . . 4  |-  ( ( 4  e.  RR  /\  C  e.  RR )  ->  ( 4  x.  C
)  e.  RR )
5250, 47, 51sylancr 663 . . 3  |-  ( ph  ->  ( 4  x.  C
)  e.  RR )
5311, 52readdcld 9626 . 2  |-  ( ph  ->  ( ( vol* `  E )  +  ( 4  x.  C ) )  e.  RR )
54 uniioombl.m . . . 4  |-  ( ph  ->  M  e.  NN )
55 uniioombl.m2 . . . 4  |-  ( ph  ->  ( abs `  (
( T `  M
)  -  sup ( ran  T ,  RR* ,  <  ) ) )  <  C
)
5626, 27, 28, 29, 11, 30, 4, 3, 31, 32, 54, 55, 20uniioombllem3 21867 . . 3  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5717, 49, 56ltled 9736 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5811, 48readdcld 9626 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( C  +  C ) )  e.  RR )
5940, 47readdcld 9626 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  e.  RR )
60 inss1 3703 . . . . . . . . . 10  |-  ( K  i^i  L )  C_  K
6160a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( K  i^i  L
)  C_  K )
62 ovolsscl 21770 . . . . . . . . 9  |-  ( ( ( K  i^i  L
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  i^i  L ) )  e.  RR )
6361, 25, 40, 62syl3anc 1229 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  RR )
6463, 47readdcld 9626 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  ( K  i^i  L
) )  +  C
)  e.  RR )
65 difssd 3617 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  K )
66 ovolsscl 21770 . . . . . . . 8  |-  ( ( ( K  \  L
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  L ) )  e.  RR )
6765, 25, 40, 66syl3anc 1229 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  \  L ) )  e.  RR )
68 uniioombl.n . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
69 uniioombl.n2 . . . . . . . 8  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( abs `  ( sum_ i  e.  ( 1 ... N ) ( vol* `  (
( (,) `  ( F `  i )
)  i^i  ( (,) `  ( G `  j
) ) ) )  -  ( vol* `  ( ( (,) `  ( G `  j )
)  i^i  A )
) ) )  < 
( C  /  M
) )
70 uniioombl.l . . . . . . . 8  |-  L  = 
U. ( ( (,) 
o.  F ) "
( 1 ... N
) )
7126, 27, 28, 29, 11, 30, 4, 3, 31, 32, 54, 55, 20, 68, 69, 70uniioombllem4 21868 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  <_  ( ( vol* `  ( K  i^i  L ) )  +  C ) )
72 imassrn 5338 . . . . . . . . . . 11  |-  ( ( (,)  o.  F )
" ( 1 ... N ) )  C_  ran  ( (,)  o.  F
)
7372unissi 4257 . . . . . . . . . 10  |-  U. (
( (,)  o.  F
) " ( 1 ... N ) ) 
C_  U. ran  ( (,) 
o.  F )
7473, 70, 293sstr4i 3528 . . . . . . . . 9  |-  L  C_  A
75 sscon 3623 . . . . . . . . 9  |-  ( L 
C_  A  ->  ( K  \  A )  C_  ( K  \  L ) )
7674, 75mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( K  \  A
)  C_  ( K  \  L ) )
7765, 25sstrd 3499 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  RR )
78 ovolss 21769 . . . . . . . 8  |-  ( ( ( K  \  A
)  C_  ( K  \  L )  /\  ( K  \  L )  C_  RR )  ->  ( vol* `  ( K  \  A ) )  <_ 
( vol* `  ( K  \  L ) ) )
7976, 77, 78syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  \  A ) )  <_  ( vol* `  ( K  \  L ) ) )
8042, 45, 64, 67, 71, 79le2addd 10176 . . . . . 6  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L
) ) ) )
8163recnd 9625 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  CC )
8247recnd 9625 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
8367recnd 9625 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  \  L ) )  e.  CC )
8481, 82, 83add32d 9807 . . . . . . 7  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
85 ioof 11631 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
86 inss2 3704 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
87 rexpssxrxp 9641 . . . . . . . . . . . . . . 15  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
8886, 87sstri 3498 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
89 fss 5729 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
9026, 88, 89sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
91 fco 5731 . . . . . . . . . . . . 13  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
9285, 90, 91sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  ( (,)  o.  F
) : NN --> ~P RR )
93 ffun 5723 . . . . . . . . . . . 12  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  Fun  ( (,)  o.  F ) )
94 funiunfv 6145 . . . . . . . . . . . 12  |-  ( Fun  ( (,)  o.  F
)  ->  U_ n  e.  ( 1 ... N
) ( ( (,) 
o.  F ) `  n )  =  U. ( ( (,)  o.  F ) " (
1 ... N ) ) )
9592, 93, 943syl 20 . . . . . . . . . . 11  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  =  U. (
( (,)  o.  F
) " ( 1 ... N ) ) )
9695, 70syl6eqr 2502 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  =  L )
97 fzfid 12062 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
98 elfznn 11723 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
99 fvco3 5935 . . . . . . . . . . . . . . 15  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
10026, 98, 99syl2an 477 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
101 ffvelrn 6014 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
10226, 98, 101syl2an 477 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
10386, 102sseldi 3487 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
104 1st2nd2 6822 . . . . . . . . . . . . . . . . 17  |-  ( ( F `  n )  e.  ( RR  X.  RR )  ->  ( F `
 n )  = 
<. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
105103, 104syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
106105fveq2d 5860 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
107 df-ov 6284 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
108106, 107syl6eqr 2502 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
109100, 108eqtrd 2484 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
110 ioombl 21848 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  e.  dom  vol
111109, 110syl6eqel 2539 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  e.  dom  vol )
112111ralrimiva 2857 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  e.  dom  vol )
113 finiunmbl 21827 . . . . . . . . . . 11  |-  ( ( ( 1 ... N
)  e.  Fin  /\  A. n  e.  ( 1 ... N ) ( ( (,)  o.  F
) `  n )  e.  dom  vol )  ->  U_ n  e.  (
1 ... N ) ( ( (,)  o.  F
) `  n )  e.  dom  vol )
11497, 112, 113syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  e.  dom  vol )
11596, 114eqeltrrd 2532 . . . . . . . . 9  |-  ( ph  ->  L  e.  dom  vol )
116 mblsplit 21816 . . . . . . . . 9  |-  ( ( L  e.  dom  vol  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  K )  =  ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) ) )
117115, 25, 40, 116syl3anc 1229 . . . . . . . 8  |-  ( ph  ->  ( vol* `  K )  =  ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) ) )
118117oveq1d 6296 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
11984, 118eqtr4d 2487 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( vol* `  K
)  +  C ) )
12080, 119breqtrd 4461 . . . . 5  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  K )  +  C ) )
12111, 47readdcld 9626 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR )
12231ovollb 21763 . . . . . . . . 9  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  K  C_ 
U. ran  ( (,)  o.  G ) )  -> 
( vol* `  K )  <_  sup ( ran  T ,  RR* ,  <  ) )
1234, 23, 122sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( vol* `  K )  <_  sup ( ran  T ,  RR* ,  <  ) )
12440, 33, 121, 123, 32letrd 9742 . . . . . . 7  |-  ( ph  ->  ( vol* `  K )  <_  (
( vol* `  E )  +  C
) )
12540, 121, 47, 124leadd1dd 10172 . . . . . 6  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( (
( vol* `  E )  +  C
)  +  C ) )
12611recnd 9625 . . . . . . 7  |-  ( ph  ->  ( vol* `  E )  e.  CC )
127126, 82, 82addassd 9621 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  E )  +  C )  +  C
)  =  ( ( vol* `  E
)  +  ( C  +  C ) ) )
128125, 127breqtrd 4461 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
12946, 59, 58, 120, 128letrd 9742 . . . 4  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
13046, 58, 48, 129leadd1dd 10172 . . 3  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( (
( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C ) ) )
13148recnd 9625 . . . . 5  |-  ( ph  ->  ( C  +  C
)  e.  CC )
132126, 131, 131addassd 9621 . . . 4  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( ( C  +  C
)  +  ( C  +  C ) ) ) )
133 2t2e4 10691 . . . . . . 7  |-  ( 2  x.  2 )  =  4
134133oveq1i 6291 . . . . . 6  |-  ( ( 2  x.  2 )  x.  C )  =  ( 4  x.  C
)
135 2cnd 10614 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
136135, 135, 82mulassd 9622 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( 2  x.  ( 2  x.  C ) ) )
137822timesd 10787 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  =  ( C  +  C ) )
138137oveq2d 6297 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
2  x.  C ) )  =  ( 2  x.  ( C  +  C ) ) )
1391312timesd 10787 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( C  +  C )
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
140136, 138, 1393eqtrd 2488 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
141134, 140syl5eqr 2498 . . . . 5  |-  ( ph  ->  ( 4  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
142141oveq2d 6297 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( 4  x.  C ) )  =  ( ( vol* `  E
)  +  ( ( C  +  C )  +  ( C  +  C ) ) ) )
143132, 142eqtr4d 2487 . . 3  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
144130, 143breqtrd 4461 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
14517, 49, 53, 57, 144letrd 9742 1  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <_  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793    \ cdif 3458    i^i cin 3460    C_ wss 3461   ~Pcpw 3997   <.cop 4020   U.cuni 4234   U_ciun 4315  Disj wdisj 4407   class class class wbr 4437    X. cxp 4987   dom cdm 4989   ran crn 4990   "cima 4992    o. ccom 4993   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   Fincfn 7518   supcsup 7902   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   RR*cxr 9630    < clt 9631    <_ cle 9632    - cmin 9810    / cdiv 10212   NNcn 10542   2c2 10591   4c4 10593   RR+crp 11229   (,)cioo 11538   [,]cicc 11541   ...cfz 11681    seqcseq 12086   abscabs 13046   sum_csu 13487   vol*covol 21747   volcvol 21748
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 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-n0 10802  df-z 10871  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-rlim 13291  df-sum 13488  df-rest 14697  df-topgen 14718  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-top 19272  df-bases 19274  df-topon 19275  df-cmp 19760  df-ovol 21749  df-vol 21750
This theorem is referenced by:  uniioombllem6  21870
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