MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniioombllem5 Structured version   Unicode version

Theorem uniioombllem5 22081
Description: Lemma for uniioombl 22083. (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 3632 . . . . 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 22072 . . . . . . 7  |-  ( ph  ->  ( U. ran  ( (,)  o.  G )  C_  U.
ran  ( [,]  o.  G )  /\  ( vol* `  ( U. ran  ( [,]  o.  G
)  \  U. ran  ( (,)  o.  G ) ) )  =  0 ) )
65simpld 457 . . . . . 6  |-  ( ph  ->  U. ran  ( (,) 
o.  G )  C_  U.
ran  ( [,]  o.  G ) )
7 ovolficcss 21966 . . . . . . 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 3427 . . . . 5  |-  ( ph  ->  U. ran  ( (,) 
o.  G )  C_  RR )
103, 9sstrd 3427 . . . 4  |-  ( ph  ->  E  C_  RR )
11 uniioombl.e . . . 4  |-  ( ph  ->  ( vol* `  E )  e.  RR )
12 ovolsscl 21982 . . . 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 1226 . . 3  |-  ( ph  ->  ( vol* `  ( E  i^i  A ) )  e.  RR )
14 difssd 3546 . . . 4  |-  ( ph  ->  ( E  \  A
)  C_  E )
15 ovolsscl 21982 . . . 4  |-  ( ( ( E  \  A
)  C_  E  /\  E  C_  RR  /\  ( vol* `  E )  e.  RR )  -> 
( vol* `  ( E  \  A ) )  e.  RR )
1614, 10, 11, 15syl3anc 1226 . . 3  |-  ( ph  ->  ( vol* `  ( E  \  A ) )  e.  RR )
1713, 16readdcld 9534 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  e.  RR )
18 inss1 3632 . . . . . 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 5260 . . . . . . . . 9  |-  ( ( (,)  o.  G )
" ( 1 ... M ) )  C_  ran  ( (,)  o.  G
)
2221unissi 4186 . . . . . . . 8  |-  U. (
( (,)  o.  G
) " ( 1 ... M ) ) 
C_  U. ran  ( (,) 
o.  G )
2320, 22eqsstri 3447 . . . . . . 7  |-  K  C_  U.
ran  ( (,)  o.  G )
2423a1i 11 . . . . . 6  |-  ( ph  ->  K  C_  U. ran  ( (,)  o.  G ) )
2524, 9sstrd 3427 . . . . 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 22075 . . . . . . 7  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
34 ssid 3436 . . . . . . . 8  |-  U. ran  ( (,)  o.  G ) 
C_  U. ran  ( (,) 
o.  G )
3531ovollb 21975 . . . . . . . 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 660 . . . . . . 7  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  <_  sup ( ran  T ,  RR* ,  <  ) )
37 ovollecl 21979 . . . . . . 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 1226 . . . . . 6  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  e.  RR )
39 ovolsscl 21982 . . . . . 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 1226 . . . . 5  |-  ( ph  ->  ( vol* `  K )  e.  RR )
41 ovolsscl 21982 . . . . 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 1226 . . . 4  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  e.  RR )
43 difssd 3546 . . . . 5  |-  ( ph  ->  ( K  \  A
)  C_  K )
44 ovolsscl 21982 . . . . 5  |-  ( ( ( K  \  A
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  A ) )  e.  RR )
4543, 25, 40, 44syl3anc 1226 . . . 4  |-  ( ph  ->  ( vol* `  ( K  \  A ) )  e.  RR )
4642, 45readdcld 9534 . . 3  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  e.  RR )
4730rpred 11177 . . . 4  |-  ( ph  ->  C  e.  RR )
4847, 47readdcld 9534 . . 3  |-  ( ph  ->  ( C  +  C
)  e.  RR )
4946, 48readdcld 9534 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  e.  RR )
50 4re 10529 . . . 4  |-  4  e.  RR
51 remulcl 9488 . . . 4  |-  ( ( 4  e.  RR  /\  C  e.  RR )  ->  ( 4  x.  C
)  e.  RR )
5250, 47, 51sylancr 661 . . 3  |-  ( ph  ->  ( 4  x.  C
)  e.  RR )
5311, 52readdcld 9534 . 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 22079 . . 3  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5717, 49, 56ltled 9644 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5811, 48readdcld 9534 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( C  +  C ) )  e.  RR )
5940, 47readdcld 9534 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  e.  RR )
60 inss1 3632 . . . . . . . . . 10  |-  ( K  i^i  L )  C_  K
6160a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( K  i^i  L
)  C_  K )
62 ovolsscl 21982 . . . . . . . . 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 1226 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  RR )
6463, 47readdcld 9534 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  ( K  i^i  L
) )  +  C
)  e.  RR )
65 difssd 3546 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  K )
66 ovolsscl 21982 . . . . . . . 8  |-  ( ( ( K  \  L
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  L ) )  e.  RR )
6765, 25, 40, 66syl3anc 1226 . . . . . . 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 22080 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  <_  ( ( vol* `  ( K  i^i  L ) )  +  C ) )
72 imassrn 5260 . . . . . . . . . . 11  |-  ( ( (,)  o.  F )
" ( 1 ... N ) )  C_  ran  ( (,)  o.  F
)
7372unissi 4186 . . . . . . . . . 10  |-  U. (
( (,)  o.  F
) " ( 1 ... N ) ) 
C_  U. ran  ( (,) 
o.  F )
7473, 70, 293sstr4i 3456 . . . . . . . . 9  |-  L  C_  A
75 sscon 3552 . . . . . . . . 9  |-  ( L 
C_  A  ->  ( K  \  A )  C_  ( K  \  L ) )
7674, 75mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( K  \  A
)  C_  ( K  \  L ) )
7765, 25sstrd 3427 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  RR )
78 ovolss 21981 . . . . . . . 8  |-  ( ( ( K  \  A
)  C_  ( K  \  L )  /\  ( K  \  L )  C_  RR )  ->  ( vol* `  ( K  \  A ) )  <_ 
( vol* `  ( K  \  L ) ) )
7976, 77, 78syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  \  A ) )  <_  ( vol* `  ( K  \  L ) ) )
8042, 45, 64, 67, 71, 79le2addd 10087 . . . . . 6  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L
) ) ) )
8163recnd 9533 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  CC )
8247recnd 9533 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
8367recnd 9533 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  \  L ) )  e.  CC )
8481, 82, 83add32d 9715 . . . . . . 7  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
85 ioof 11543 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
86 inss2 3633 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
87 rexpssxrxp 9549 . . . . . . . . . . . . . . 15  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
8886, 87sstri 3426 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
89 fss 5647 . . . . . . . . . . . . . 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 660 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
91 fco 5649 . . . . . . . . . . . . 13  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
9285, 90, 91sylancr 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( (,)  o.  F
) : NN --> ~P RR )
93 ffun 5641 . . . . . . . . . . . 12  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  Fun  ( (,)  o.  F ) )
94 funiunfv 6061 . . . . . . . . . . . 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 2441 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  =  L )
97 fzfid 11986 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
98 elfznn 11635 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
99 fvco3 5851 . . . . . . . . . . . . . . 15  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
10026, 98, 99syl2an 475 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
101 ffvelrn 5931 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
10226, 98, 101syl2an 475 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
10386, 102sseldi 3415 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
104 1st2nd2 6736 . . . . . . . . . . . . . . . . 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 5778 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
107 df-ov 6199 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
108106, 107syl6eqr 2441 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
109100, 108eqtrd 2423 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
110 ioombl 22060 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  e.  dom  vol
111109, 110syl6eqel 2478 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  e.  dom  vol )
112111ralrimiva 2796 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  e.  dom  vol )
113 finiunmbl 22039 . . . . . . . . . . 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 659 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  e.  dom  vol )
11596, 114eqeltrrd 2471 . . . . . . . . 9  |-  ( ph  ->  L  e.  dom  vol )
116 mblsplit 22028 . . . . . . . . 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 1226 . . . . . . . 8  |-  ( ph  ->  ( vol* `  K )  =  ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) ) )
118117oveq1d 6211 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
11984, 118eqtr4d 2426 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( vol* `  K
)  +  C ) )
12080, 119breqtrd 4391 . . . . 5  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  K )  +  C ) )
12111, 47readdcld 9534 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR )
12231ovollb 21975 . . . . . . . . 9  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  K  C_ 
U. ran  ( (,)  o.  G ) )  -> 
( vol* `  K )  <_  sup ( ran  T ,  RR* ,  <  ) )
1234, 23, 122sylancl 660 . . . . . . . 8  |-  ( ph  ->  ( vol* `  K )  <_  sup ( ran  T ,  RR* ,  <  ) )
12440, 33, 121, 123, 32letrd 9650 . . . . . . 7  |-  ( ph  ->  ( vol* `  K )  <_  (
( vol* `  E )  +  C
) )
12540, 121, 47, 124leadd1dd 10083 . . . . . 6  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( (
( vol* `  E )  +  C
)  +  C ) )
12611recnd 9533 . . . . . . 7  |-  ( ph  ->  ( vol* `  E )  e.  CC )
127126, 82, 82addassd 9529 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  E )  +  C )  +  C
)  =  ( ( vol* `  E
)  +  ( C  +  C ) ) )
128125, 127breqtrd 4391 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
12946, 59, 58, 120, 128letrd 9650 . . . 4  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
13046, 58, 48, 129leadd1dd 10083 . . 3  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( (
( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C ) ) )
13148recnd 9533 . . . . 5  |-  ( ph  ->  ( C  +  C
)  e.  CC )
132126, 131, 131addassd 9529 . . . 4  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( ( C  +  C
)  +  ( C  +  C ) ) ) )
133 2t2e4 10602 . . . . . . 7  |-  ( 2  x.  2 )  =  4
134133oveq1i 6206 . . . . . 6  |-  ( ( 2  x.  2 )  x.  C )  =  ( 4  x.  C
)
135 2cnd 10525 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
136135, 135, 82mulassd 9530 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( 2  x.  ( 2  x.  C ) ) )
137822timesd 10698 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  =  ( C  +  C ) )
138137oveq2d 6212 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
2  x.  C ) )  =  ( 2  x.  ( C  +  C ) ) )
1391312timesd 10698 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( C  +  C )
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
140136, 138, 1393eqtrd 2427 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
141134, 140syl5eqr 2437 . . . . 5  |-  ( ph  ->  ( 4  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
142141oveq2d 6212 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( 4  x.  C ) )  =  ( ( vol* `  E
)  +  ( ( C  +  C )  +  ( C  +  C ) ) ) )
143132, 142eqtr4d 2426 . . 3  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
144130, 143breqtrd 4391 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
14517, 49, 53, 57, 144letrd 9650 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 367    = wceq 1399    e. wcel 1826   A.wral 2732    \ cdif 3386    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   <.cop 3950   U.cuni 4163   U_ciun 4243  Disj wdisj 4338   class class class wbr 4367    X. cxp 4911   dom cdm 4913   ran crn 4914   "cima 4916    o. ccom 4917   Fun wfun 5490   -->wf 5492   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698   Fincfn 7435   supcsup 7815   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408   RR*cxr 9538    < clt 9539    <_ cle 9540    - cmin 9718    / cdiv 10123   NNcn 10452   2c2 10502   4c4 10504   RR+crp 11139   (,)cioo 11450   [,]cicc 11453   ...cfz 11593    seqcseq 12010   abscabs 13069   sum_csu 13510   vol*covol 21959   volcvol 21960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-disj 4339  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-acn 8236  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-rlim 13314  df-sum 13511  df-rest 14830  df-topgen 14851  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-top 19484  df-bases 19486  df-topon 19487  df-cmp 19973  df-ovol 21961  df-vol 21962
This theorem is referenced by:  uniioombllem6  22082
  Copyright terms: Public domain W3C validator