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Theorem uniioombllem5 21067
Description: Lemma for uniioombl 21069. (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 3570 . . . . 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 21058 . . . . . . 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 20953 . . . . . . 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 3366 . . . . 5  |-  ( ph  ->  U. ran  ( (,) 
o.  G )  C_  RR )
103, 9sstrd 3366 . . . 4  |-  ( ph  ->  E  C_  RR )
11 uniioombl.e . . . 4  |-  ( ph  ->  ( vol* `  E )  e.  RR )
12 ovolsscl 20969 . . . 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 1218 . . 3  |-  ( ph  ->  ( vol* `  ( E  i^i  A ) )  e.  RR )
14 difssd 3484 . . . 4  |-  ( ph  ->  ( E  \  A
)  C_  E )
15 ovolsscl 20969 . . . 4  |-  ( ( ( E  \  A
)  C_  E  /\  E  C_  RR  /\  ( vol* `  E )  e.  RR )  -> 
( vol* `  ( E  \  A ) )  e.  RR )
1614, 10, 11, 15syl3anc 1218 . . 3  |-  ( ph  ->  ( vol* `  ( E  \  A ) )  e.  RR )
1713, 16readdcld 9413 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  e.  RR )
18 inss1 3570 . . . . . 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 5180 . . . . . . . . 9  |-  ( ( (,)  o.  G )
" ( 1 ... M ) )  C_  ran  ( (,)  o.  G
)
2221unissi 4114 . . . . . . . 8  |-  U. (
( (,)  o.  G
) " ( 1 ... M ) ) 
C_  U. ran  ( (,) 
o.  G )
2320, 22eqsstri 3386 . . . . . . 7  |-  K  C_  U.
ran  ( (,)  o.  G )
2423a1i 11 . . . . . 6  |-  ( ph  ->  K  C_  U. ran  ( (,)  o.  G ) )
2524, 9sstrd 3366 . . . . 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 21061 . . . . . . 7  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
34 ssid 3375 . . . . . . . 8  |-  U. ran  ( (,)  o.  G ) 
C_  U. ran  ( (,) 
o.  G )
3531ovollb 20962 . . . . . . . 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 20966 . . . . . . 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 1218 . . . . . 6  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  e.  RR )
39 ovolsscl 20969 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( vol* `  K )  e.  RR )
41 ovolsscl 20969 . . . . 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 1218 . . . 4  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  e.  RR )
43 difssd 3484 . . . . 5  |-  ( ph  ->  ( K  \  A
)  C_  K )
44 ovolsscl 20969 . . . . 5  |-  ( ( ( K  \  A
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  A ) )  e.  RR )
4543, 25, 40, 44syl3anc 1218 . . . 4  |-  ( ph  ->  ( vol* `  ( K  \  A ) )  e.  RR )
4642, 45readdcld 9413 . . 3  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  e.  RR )
4730rpred 11027 . . . 4  |-  ( ph  ->  C  e.  RR )
4847, 47readdcld 9413 . . 3  |-  ( ph  ->  ( C  +  C
)  e.  RR )
4946, 48readdcld 9413 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  e.  RR )
50 4re 10398 . . . 4  |-  4  e.  RR
51 remulcl 9367 . . . 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 9413 . 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 21065 . . 3  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5717, 49, 56ltled 9522 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5811, 48readdcld 9413 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( C  +  C ) )  e.  RR )
5940, 47readdcld 9413 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  e.  RR )
60 inss1 3570 . . . . . . . . . 10  |-  ( K  i^i  L )  C_  K
6160a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( K  i^i  L
)  C_  K )
62 ovolsscl 20969 . . . . . . . . 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 1218 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  RR )
6463, 47readdcld 9413 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  ( K  i^i  L
) )  +  C
)  e.  RR )
65 difssd 3484 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  K )
66 ovolsscl 20969 . . . . . . . 8  |-  ( ( ( K  \  L
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  L ) )  e.  RR )
6765, 25, 40, 66syl3anc 1218 . . . . . . 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 21066 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  <_  ( ( vol* `  ( K  i^i  L ) )  +  C ) )
72 imassrn 5180 . . . . . . . . . . 11  |-  ( ( (,)  o.  F )
" ( 1 ... N ) )  C_  ran  ( (,)  o.  F
)
7372unissi 4114 . . . . . . . . . 10  |-  U. (
( (,)  o.  F
) " ( 1 ... N ) ) 
C_  U. ran  ( (,) 
o.  F )
7473, 70, 293sstr4i 3395 . . . . . . . . 9  |-  L  C_  A
75 sscon 3490 . . . . . . . . 9  |-  ( L 
C_  A  ->  ( K  \  A )  C_  ( K  \  L ) )
7674, 75mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( K  \  A
)  C_  ( K  \  L ) )
7765, 25sstrd 3366 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  RR )
78 ovolss 20968 . . . . . . . 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 9957 . . . . . 6  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L
) ) ) )
8163recnd 9412 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  CC )
8247recnd 9412 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
8367recnd 9412 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  \  L ) )  e.  CC )
8481, 82, 83add32d 9592 . . . . . . 7  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
85 ioof 11387 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
86 inss2 3571 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
87 rexpssxrxp 9428 . . . . . . . . . . . . . . 15  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
8886, 87sstri 3365 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
89 fss 5567 . . . . . . . . . . . . . 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 5568 . . . . . . . . . . . . 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 5561 . . . . . . . . . . . 12  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  Fun  ( (,)  o.  F ) )
94 funiunfv 5965 . . . . . . . . . . . 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 2493 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  =  L )
97 fzfid 11795 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
98 elfznn 11478 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
99 fvco3 5768 . . . . . . . . . . . . . . 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 5841 . . . . . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
104 1st2nd2 6613 . . . . . . . . . . . . . . . . 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 5695 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
107 df-ov 6094 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
108106, 107syl6eqr 2493 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
109100, 108eqtrd 2475 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
110 ioombl 21046 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  e.  dom  vol
111109, 110syl6eqel 2531 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  e.  dom  vol )
112111ralrimiva 2799 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  e.  dom  vol )
113 finiunmbl 21025 . . . . . . . . . . 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 2518 . . . . . . . . 9  |-  ( ph  ->  L  e.  dom  vol )
116 mblsplit 21015 . . . . . . . . 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 1218 . . . . . . . 8  |-  ( ph  ->  ( vol* `  K )  =  ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) ) )
118117oveq1d 6106 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
11984, 118eqtr4d 2478 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( vol* `  K
)  +  C ) )
12080, 119breqtrd 4316 . . . . 5  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  K )  +  C ) )
12111, 47readdcld 9413 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR )
12231ovollb 20962 . . . . . . . . 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 9528 . . . . . . 7  |-  ( ph  ->  ( vol* `  K )  <_  (
( vol* `  E )  +  C
) )
12540, 121, 47, 124leadd1dd 9953 . . . . . 6  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( (
( vol* `  E )  +  C
)  +  C ) )
12611recnd 9412 . . . . . . 7  |-  ( ph  ->  ( vol* `  E )  e.  CC )
127126, 82, 82addassd 9408 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  E )  +  C )  +  C
)  =  ( ( vol* `  E
)  +  ( C  +  C ) ) )
128125, 127breqtrd 4316 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
12946, 59, 58, 120, 128letrd 9528 . . . 4  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
13046, 58, 48, 129leadd1dd 9953 . . 3  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( (
( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C ) ) )
13148recnd 9412 . . . . 5  |-  ( ph  ->  ( C  +  C
)  e.  CC )
132126, 131, 131addassd 9408 . . . 4  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( ( C  +  C
)  +  ( C  +  C ) ) ) )
133 2t2e4 10471 . . . . . . 7  |-  ( 2  x.  2 )  =  4
134133oveq1i 6101 . . . . . 6  |-  ( ( 2  x.  2 )  x.  C )  =  ( 4  x.  C
)
135 2cnd 10394 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
136135, 135, 82mulassd 9409 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( 2  x.  ( 2  x.  C ) ) )
137822timesd 10567 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  =  ( C  +  C ) )
138137oveq2d 6107 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
2  x.  C ) )  =  ( 2  x.  ( C  +  C ) ) )
1391312timesd 10567 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( C  +  C )
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
140136, 138, 1393eqtrd 2479 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
141134, 140syl5eqr 2489 . . . . 5  |-  ( ph  ->  ( 4  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
142141oveq2d 6107 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( 4  x.  C ) )  =  ( ( vol* `  E
)  +  ( ( C  +  C )  +  ( C  +  C ) ) ) )
143132, 142eqtr4d 2478 . . 3  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
144130, 143breqtrd 4316 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
14517, 49, 53, 57, 144letrd 9528 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 1369    e. wcel 1756   A.wral 2715    \ cdif 3325    i^i cin 3327    C_ wss 3328   ~Pcpw 3860   <.cop 3883   U.cuni 4091   U_ciun 4171  Disj wdisj 4262   class class class wbr 4292    X. cxp 4838   dom cdm 4840   ran crn 4841   "cima 4843    o. ccom 4844   Fun wfun 5412   -->wf 5414   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   Fincfn 7310   supcsup 7690   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287   RR*cxr 9417    < clt 9418    <_ cle 9419    - cmin 9595    / cdiv 9993   NNcn 10322   2c2 10371   4c4 10373   RR+crp 10991   (,)cioo 11300   [,]cicc 11303   ...cfz 11437    seqcseq 11806   abscabs 12723   sum_csu 13163   vol*covol 20946   volcvol 20947
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-disj 4263  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-acn 8112  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-rest 14361  df-topgen 14382  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-top 18503  df-bases 18505  df-topon 18506  df-cmp 18990  df-ovol 20948  df-vol 20949
This theorem is referenced by:  uniioombllem6  21068
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