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Theorem uniioombllem5 21759
Description: Lemma for uniioombl 21761. (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 3718 . . . . 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 21750 . . . . . . 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 21644 . . . . . . 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 3514 . . . . 5  |-  ( ph  ->  U. ran  ( (,) 
o.  G )  C_  RR )
103, 9sstrd 3514 . . . 4  |-  ( ph  ->  E  C_  RR )
11 uniioombl.e . . . 4  |-  ( ph  ->  ( vol* `  E )  e.  RR )
12 ovolsscl 21660 . . . 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 1228 . . 3  |-  ( ph  ->  ( vol* `  ( E  i^i  A ) )  e.  RR )
14 difssd 3632 . . . 4  |-  ( ph  ->  ( E  \  A
)  C_  E )
15 ovolsscl 21660 . . . 4  |-  ( ( ( E  \  A
)  C_  E  /\  E  C_  RR  /\  ( vol* `  E )  e.  RR )  -> 
( vol* `  ( E  \  A ) )  e.  RR )
1614, 10, 11, 15syl3anc 1228 . . 3  |-  ( ph  ->  ( vol* `  ( E  \  A ) )  e.  RR )
1713, 16readdcld 9623 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  e.  RR )
18 inss1 3718 . . . . . 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 5348 . . . . . . . . 9  |-  ( ( (,)  o.  G )
" ( 1 ... M ) )  C_  ran  ( (,)  o.  G
)
2221unissi 4268 . . . . . . . 8  |-  U. (
( (,)  o.  G
) " ( 1 ... M ) ) 
C_  U. ran  ( (,) 
o.  G )
2320, 22eqsstri 3534 . . . . . . 7  |-  K  C_  U.
ran  ( (,)  o.  G )
2423a1i 11 . . . . . 6  |-  ( ph  ->  K  C_  U. ran  ( (,)  o.  G ) )
2524, 9sstrd 3514 . . . . 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 21753 . . . . . . 7  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
34 ssid 3523 . . . . . . . 8  |-  U. ran  ( (,)  o.  G ) 
C_  U. ran  ( (,) 
o.  G )
3531ovollb 21653 . . . . . . . 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 21657 . . . . . . 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 1228 . . . . . 6  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  e.  RR )
39 ovolsscl 21660 . . . . . 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 1228 . . . . 5  |-  ( ph  ->  ( vol* `  K )  e.  RR )
41 ovolsscl 21660 . . . . 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 1228 . . . 4  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  e.  RR )
43 difssd 3632 . . . . 5  |-  ( ph  ->  ( K  \  A
)  C_  K )
44 ovolsscl 21660 . . . . 5  |-  ( ( ( K  \  A
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  A ) )  e.  RR )
4543, 25, 40, 44syl3anc 1228 . . . 4  |-  ( ph  ->  ( vol* `  ( K  \  A ) )  e.  RR )
4642, 45readdcld 9623 . . 3  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  e.  RR )
4730rpred 11256 . . . 4  |-  ( ph  ->  C  e.  RR )
4847, 47readdcld 9623 . . 3  |-  ( ph  ->  ( C  +  C
)  e.  RR )
4946, 48readdcld 9623 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  e.  RR )
50 4re 10612 . . . 4  |-  4  e.  RR
51 remulcl 9577 . . . 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 9623 . 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 21757 . . 3  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5717, 49, 56ltled 9732 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5811, 48readdcld 9623 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( C  +  C ) )  e.  RR )
5940, 47readdcld 9623 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  e.  RR )
60 inss1 3718 . . . . . . . . . 10  |-  ( K  i^i  L )  C_  K
6160a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( K  i^i  L
)  C_  K )
62 ovolsscl 21660 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  RR )
6463, 47readdcld 9623 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  ( K  i^i  L
) )  +  C
)  e.  RR )
65 difssd 3632 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  K )
66 ovolsscl 21660 . . . . . . . 8  |-  ( ( ( K  \  L
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  L ) )  e.  RR )
6765, 25, 40, 66syl3anc 1228 . . . . . . 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 21758 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  <_  ( ( vol* `  ( K  i^i  L ) )  +  C ) )
72 imassrn 5348 . . . . . . . . . . 11  |-  ( ( (,)  o.  F )
" ( 1 ... N ) )  C_  ran  ( (,)  o.  F
)
7372unissi 4268 . . . . . . . . . 10  |-  U. (
( (,)  o.  F
) " ( 1 ... N ) ) 
C_  U. ran  ( (,) 
o.  F )
7473, 70, 293sstr4i 3543 . . . . . . . . 9  |-  L  C_  A
75 sscon 3638 . . . . . . . . 9  |-  ( L 
C_  A  ->  ( K  \  A )  C_  ( K  \  L ) )
7674, 75mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( K  \  A
)  C_  ( K  \  L ) )
7765, 25sstrd 3514 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  RR )
78 ovolss 21659 . . . . . . . 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 10170 . . . . . 6  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L
) ) ) )
8163recnd 9622 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  CC )
8247recnd 9622 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
8367recnd 9622 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  \  L ) )  e.  CC )
8481, 82, 83add32d 9802 . . . . . . 7  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
85 ioof 11622 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
86 inss2 3719 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
87 rexpssxrxp 9638 . . . . . . . . . . . . . . 15  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
8886, 87sstri 3513 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
89 fss 5739 . . . . . . . . . . . . . 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 5741 . . . . . . . . . . . . 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 5733 . . . . . . . . . . . 12  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  Fun  ( (,)  o.  F ) )
94 funiunfv 6148 . . . . . . . . . . . 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 2526 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  =  L )
97 fzfid 12051 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
98 elfznn 11714 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
99 fvco3 5944 . . . . . . . . . . . . . . 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 6019 . . . . . . . . . . . . . . . . . . 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 3502 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
104 1st2nd2 6821 . . . . . . . . . . . . . . . . 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 5870 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
107 df-ov 6287 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
108106, 107syl6eqr 2526 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
109100, 108eqtrd 2508 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
110 ioombl 21738 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  e.  dom  vol
111109, 110syl6eqel 2563 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  e.  dom  vol )
112111ralrimiva 2878 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  e.  dom  vol )
113 finiunmbl 21717 . . . . . . . . . . 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 2556 . . . . . . . . 9  |-  ( ph  ->  L  e.  dom  vol )
116 mblsplit 21706 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( ph  ->  ( vol* `  K )  =  ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) ) )
118117oveq1d 6299 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
11984, 118eqtr4d 2511 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( vol* `  K
)  +  C ) )
12080, 119breqtrd 4471 . . . . 5  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  K )  +  C ) )
12111, 47readdcld 9623 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR )
12231ovollb 21653 . . . . . . . . 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 9738 . . . . . . 7  |-  ( ph  ->  ( vol* `  K )  <_  (
( vol* `  E )  +  C
) )
12540, 121, 47, 124leadd1dd 10166 . . . . . 6  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( (
( vol* `  E )  +  C
)  +  C ) )
12611recnd 9622 . . . . . . 7  |-  ( ph  ->  ( vol* `  E )  e.  CC )
127126, 82, 82addassd 9618 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  E )  +  C )  +  C
)  =  ( ( vol* `  E
)  +  ( C  +  C ) ) )
128125, 127breqtrd 4471 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
12946, 59, 58, 120, 128letrd 9738 . . . 4  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
13046, 58, 48, 129leadd1dd 10166 . . 3  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( (
( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C ) ) )
13148recnd 9622 . . . . 5  |-  ( ph  ->  ( C  +  C
)  e.  CC )
132126, 131, 131addassd 9618 . . . 4  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( ( C  +  C
)  +  ( C  +  C ) ) ) )
133 2t2e4 10685 . . . . . . 7  |-  ( 2  x.  2 )  =  4
134133oveq1i 6294 . . . . . 6  |-  ( ( 2  x.  2 )  x.  C )  =  ( 4  x.  C
)
135 2cnd 10608 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
136135, 135, 82mulassd 9619 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( 2  x.  ( 2  x.  C ) ) )
137822timesd 10781 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  =  ( C  +  C ) )
138137oveq2d 6300 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
2  x.  C ) )  =  ( 2  x.  ( C  +  C ) ) )
1391312timesd 10781 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( C  +  C )
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
140136, 138, 1393eqtrd 2512 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
141134, 140syl5eqr 2522 . . . . 5  |-  ( ph  ->  ( 4  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
142141oveq2d 6300 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( 4  x.  C ) )  =  ( ( vol* `  E
)  +  ( ( C  +  C )  +  ( C  +  C ) ) ) )
143132, 142eqtr4d 2511 . . 3  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
144130, 143breqtrd 4471 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
14517, 49, 53, 57, 144letrd 9738 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 1379    e. wcel 1767   A.wral 2814    \ cdif 3473    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   <.cop 4033   U.cuni 4245   U_ciun 4325  Disj wdisj 4417   class class class wbr 4447    X. cxp 4997   dom cdm 4999   ran crn 5000   "cima 5002    o. ccom 5003   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   Fincfn 7516   supcsup 7900   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497   RR*cxr 9627    < clt 9628    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   2c2 10585   4c4 10587   RR+crp 11220   (,)cioo 11529   [,]cicc 11532   ...cfz 11672    seqcseq 12075   abscabs 13030   sum_csu 13471   vol*covol 21637   volcvol 21638
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-rest 14678  df-topgen 14699  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-top 19194  df-bases 19196  df-topon 19197  df-cmp 19681  df-ovol 21639  df-vol 21640
This theorem is referenced by:  uniioombllem6  21760
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