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

Theorem uniioombllem5 20967
Description: Lemma for uniioombl 20969. (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 3567 . . . . 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 20958 . . . . . . 7  |-  ( ph  ->  ( U. ran  ( (,)  o.  G )  C_  U.
ran  ( [,]  o.  G )  /\  ( vol* `  ( U. ran  ( [,]  o.  G
)  \  U. ran  ( (,)  o.  G ) ) )  =  0 ) )
65simpld 456 . . . . . 6  |-  ( ph  ->  U. ran  ( (,) 
o.  G )  C_  U.
ran  ( [,]  o.  G ) )
7 ovolficcss 20853 . . . . . . 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 3363 . . . . 5  |-  ( ph  ->  U. ran  ( (,) 
o.  G )  C_  RR )
103, 9sstrd 3363 . . . 4  |-  ( ph  ->  E  C_  RR )
11 uniioombl.e . . . 4  |-  ( ph  ->  ( vol* `  E )  e.  RR )
12 ovolsscl 20869 . . . 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 1213 . . 3  |-  ( ph  ->  ( vol* `  ( E  i^i  A ) )  e.  RR )
14 difssd 3481 . . . 4  |-  ( ph  ->  ( E  \  A
)  C_  E )
15 ovolsscl 20869 . . . 4  |-  ( ( ( E  \  A
)  C_  E  /\  E  C_  RR  /\  ( vol* `  E )  e.  RR )  -> 
( vol* `  ( E  \  A ) )  e.  RR )
1614, 10, 11, 15syl3anc 1213 . . 3  |-  ( ph  ->  ( vol* `  ( E  \  A ) )  e.  RR )
1713, 16readdcld 9409 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  e.  RR )
18 inss1 3567 . . . . . 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 5177 . . . . . . . . 9  |-  ( ( (,)  o.  G )
" ( 1 ... M ) )  C_  ran  ( (,)  o.  G
)
2221unissi 4111 . . . . . . . 8  |-  U. (
( (,)  o.  G
) " ( 1 ... M ) ) 
C_  U. ran  ( (,) 
o.  G )
2320, 22eqsstri 3383 . . . . . . 7  |-  K  C_  U.
ran  ( (,)  o.  G )
2423a1i 11 . . . . . 6  |-  ( ph  ->  K  C_  U. ran  ( (,)  o.  G ) )
2524, 9sstrd 3363 . . . . 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 20961 . . . . . . 7  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
34 ssid 3372 . . . . . . . 8  |-  U. ran  ( (,)  o.  G ) 
C_  U. ran  ( (,) 
o.  G )
3531ovollb 20862 . . . . . . . 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 657 . . . . . . 7  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  <_  sup ( ran  T ,  RR* ,  <  ) )
37 ovollecl 20866 . . . . . . 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 1213 . . . . . 6  |-  ( ph  ->  ( vol* `  U. ran  ( (,)  o.  G ) )  e.  RR )
39 ovolsscl 20869 . . . . . 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 1213 . . . . 5  |-  ( ph  ->  ( vol* `  K )  e.  RR )
41 ovolsscl 20869 . . . . 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 1213 . . . 4  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  e.  RR )
43 difssd 3481 . . . . 5  |-  ( ph  ->  ( K  \  A
)  C_  K )
44 ovolsscl 20869 . . . . 5  |-  ( ( ( K  \  A
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  A ) )  e.  RR )
4543, 25, 40, 44syl3anc 1213 . . . 4  |-  ( ph  ->  ( vol* `  ( K  \  A ) )  e.  RR )
4642, 45readdcld 9409 . . 3  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  e.  RR )
4730rpred 11023 . . . 4  |-  ( ph  ->  C  e.  RR )
4847, 47readdcld 9409 . . 3  |-  ( ph  ->  ( C  +  C
)  e.  RR )
4946, 48readdcld 9409 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  e.  RR )
50 4re 10394 . . . 4  |-  4  e.  RR
51 remulcl 9363 . . . 4  |-  ( ( 4  e.  RR  /\  C  e.  RR )  ->  ( 4  x.  C
)  e.  RR )
5250, 47, 51sylancr 658 . . 3  |-  ( ph  ->  ( 4  x.  C
)  e.  RR )
5311, 52readdcld 9409 . 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 20965 . . 3  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5717, 49, 56ltled 9518 . 2  |-  ( ph  ->  ( ( vol* `  ( E  i^i  A
) )  +  ( vol* `  ( E  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) ) )
5811, 48readdcld 9409 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( C  +  C ) )  e.  RR )
5940, 47readdcld 9409 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  e.  RR )
60 inss1 3567 . . . . . . . . . 10  |-  ( K  i^i  L )  C_  K
6160a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( K  i^i  L
)  C_  K )
62 ovolsscl 20869 . . . . . . . . 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 1213 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  RR )
6463, 47readdcld 9409 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  ( K  i^i  L
) )  +  C
)  e.  RR )
65 difssd 3481 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  K )
66 ovolsscl 20869 . . . . . . . 8  |-  ( ( ( K  \  L
)  C_  K  /\  K  C_  RR  /\  ( vol* `  K )  e.  RR )  -> 
( vol* `  ( K  \  L ) )  e.  RR )
6765, 25, 40, 66syl3anc 1213 . . . . . . 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 20966 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  i^i  A ) )  <_  ( ( vol* `  ( K  i^i  L ) )  +  C ) )
72 imassrn 5177 . . . . . . . . . . 11  |-  ( ( (,)  o.  F )
" ( 1 ... N ) )  C_  ran  ( (,)  o.  F
)
7372unissi 4111 . . . . . . . . . 10  |-  U. (
( (,)  o.  F
) " ( 1 ... N ) ) 
C_  U. ran  ( (,) 
o.  F )
7473, 70, 293sstr4i 3392 . . . . . . . . 9  |-  L  C_  A
75 sscon 3487 . . . . . . . . 9  |-  ( L 
C_  A  ->  ( K  \  A )  C_  ( K  \  L ) )
7674, 75mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( K  \  A
)  C_  ( K  \  L ) )
7765, 25sstrd 3363 . . . . . . . 8  |-  ( ph  ->  ( K  \  L
)  C_  RR )
78 ovolss 20868 . . . . . . . 8  |-  ( ( ( K  \  A
)  C_  ( K  \  L )  /\  ( K  \  L )  C_  RR )  ->  ( vol* `  ( K  \  A ) )  <_ 
( vol* `  ( K  \  L ) ) )
7976, 77, 78syl2anc 656 . . . . . . 7  |-  ( ph  ->  ( vol* `  ( K  \  A ) )  <_  ( vol* `  ( K  \  L ) ) )
8042, 45, 64, 67, 71, 79le2addd 9953 . . . . . 6  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( (
( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L
) ) ) )
8163recnd 9408 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  i^i  L ) )  e.  CC )
8247recnd 9408 . . . . . . . 8  |-  ( ph  ->  C  e.  CC )
8367recnd 9408 . . . . . . . 8  |-  ( ph  ->  ( vol* `  ( K  \  L ) )  e.  CC )
8481, 82, 83add32d 9588 . . . . . . 7  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
85 ioof 11383 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
86 inss2 3568 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
87 rexpssxrxp 9424 . . . . . . . . . . . . . . 15  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
8886, 87sstri 3362 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
89 fss 5564 . . . . . . . . . . . . . 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 657 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
91 fco 5565 . . . . . . . . . . . . 13  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
9285, 90, 91sylancr 658 . . . . . . . . . . . 12  |-  ( ph  ->  ( (,)  o.  F
) : NN --> ~P RR )
93 ffun 5558 . . . . . . . . . . . 12  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  Fun  ( (,)  o.  F ) )
94 funiunfv 5962 . . . . . . . . . . . 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 2491 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  =  L )
97 fzfid 11791 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
98 elfznn 11474 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
99 fvco3 5765 . . . . . . . . . . . . . . 15  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
10026, 98, 99syl2an 474 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
101 ffvelrn 5838 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
10226, 98, 101syl2an 474 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
10386, 102sseldi 3351 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
104 1st2nd2 6612 . . . . . . . . . . . . . . . . 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 5692 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
107 df-ov 6093 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
108106, 107syl6eqr 2491 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
109100, 108eqtrd 2473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
110 ioombl 20946 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  e.  dom  vol
111109, 110syl6eqel 2529 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( (,)  o.  F
) `  n )  e.  dom  vol )
112111ralrimiva 2797 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  e.  dom  vol )
113 finiunmbl 20925 . . . . . . . . . . 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 656 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... N ) ( ( (,)  o.  F ) `  n
)  e.  dom  vol )
11596, 114eqeltrrd 2516 . . . . . . . . 9  |-  ( ph  ->  L  e.  dom  vol )
116 mblsplit 20915 . . . . . . . . 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 1213 . . . . . . . 8  |-  ( ph  ->  ( vol* `  K )  =  ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) ) )
118117oveq1d 6105 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  =  ( ( ( vol* `  ( K  i^i  L ) )  +  ( vol* `  ( K  \  L ) ) )  +  C ) )
11984, 118eqtr4d 2476 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  L ) )  +  C )  +  ( vol* `  ( K  \  L ) ) )  =  ( ( vol* `  K
)  +  C ) )
12080, 119breqtrd 4313 . . . . 5  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  K )  +  C ) )
12111, 47readdcld 9409 . . . . . . 7  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR )
12231ovollb 20862 . . . . . . . . 9  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  K  C_ 
U. ran  ( (,)  o.  G ) )  -> 
( vol* `  K )  <_  sup ( ran  T ,  RR* ,  <  ) )
1234, 23, 122sylancl 657 . . . . . . . 8  |-  ( ph  ->  ( vol* `  K )  <_  sup ( ran  T ,  RR* ,  <  ) )
12440, 33, 121, 123, 32letrd 9524 . . . . . . 7  |-  ( ph  ->  ( vol* `  K )  <_  (
( vol* `  E )  +  C
) )
12540, 121, 47, 124leadd1dd 9949 . . . . . 6  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( (
( vol* `  E )  +  C
)  +  C ) )
12611recnd 9408 . . . . . . 7  |-  ( ph  ->  ( vol* `  E )  e.  CC )
127126, 82, 82addassd 9404 . . . . . 6  |-  ( ph  ->  ( ( ( vol* `  E )  +  C )  +  C
)  =  ( ( vol* `  E
)  +  ( C  +  C ) ) )
128125, 127breqtrd 4313 . . . . 5  |-  ( ph  ->  ( ( vol* `  K )  +  C
)  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
12946, 59, 58, 120, 128letrd 9524 . . . 4  |-  ( ph  ->  ( ( vol* `  ( K  i^i  A
) )  +  ( vol* `  ( K  \  A ) ) )  <_  ( ( vol* `  E )  +  ( C  +  C ) ) )
13046, 58, 48, 129leadd1dd 9949 . . 3  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( (
( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C ) ) )
13148recnd 9408 . . . . 5  |-  ( ph  ->  ( C  +  C
)  e.  CC )
132126, 131, 131addassd 9404 . . . 4  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( ( C  +  C
)  +  ( C  +  C ) ) ) )
133 2t2e4 10467 . . . . . . 7  |-  ( 2  x.  2 )  =  4
134133oveq1i 6100 . . . . . 6  |-  ( ( 2  x.  2 )  x.  C )  =  ( 4  x.  C
)
135 2cnd 10390 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
136135, 135, 82mulassd 9405 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( 2  x.  ( 2  x.  C ) ) )
137822timesd 10563 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  =  ( C  +  C ) )
138137oveq2d 6106 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
2  x.  C ) )  =  ( 2  x.  ( C  +  C ) ) )
1391312timesd 10563 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( C  +  C )
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
140136, 138, 1393eqtrd 2477 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  2 )  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
141134, 140syl5eqr 2487 . . . . 5  |-  ( ph  ->  ( 4  x.  C
)  =  ( ( C  +  C )  +  ( C  +  C ) ) )
142141oveq2d 6106 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  ( 4  x.  C ) )  =  ( ( vol* `  E
)  +  ( ( C  +  C )  +  ( C  +  C ) ) ) )
143132, 142eqtr4d 2476 . . 3  |-  ( ph  ->  ( ( ( vol* `  E )  +  ( C  +  C ) )  +  ( C  +  C
) )  =  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
144130, 143breqtrd 4313 . 2  |-  ( ph  ->  ( ( ( vol* `  ( K  i^i  A ) )  +  ( vol* `  ( K  \  A ) ) )  +  ( C  +  C ) )  <_  ( ( vol* `  E )  +  ( 4  x.  C ) ) )
14517, 49, 53, 57, 144letrd 9524 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 1364    e. wcel 1761   A.wral 2713    \ cdif 3322    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   <.cop 3880   U.cuni 4088   U_ciun 4168  Disj wdisj 4259   class class class wbr 4289    X. cxp 4834   dom cdm 4836   ran crn 4837   "cima 4839    o. ccom 4840   Fun wfun 5409   -->wf 5411   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   Fincfn 7306   supcsup 7686   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283   RR*cxr 9413    < clt 9414    <_ cle 9415    - cmin 9591    / cdiv 9989   NNcn 10318   2c2 10367   4c4 10369   RR+crp 10987   (,)cioo 11296   [,]cicc 11299   ...cfz 11433    seqcseq 11802   abscabs 12719   sum_csu 13159   vol*covol 20846   volcvol 20847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-rest 14357  df-topgen 14378  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-top 18403  df-bases 18405  df-topon 18406  df-cmp 18890  df-ovol 20848  df-vol 20849
This theorem is referenced by:  uniioombllem6  20968
  Copyright terms: Public domain W3C validator