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Theorem ovolshftlem1 21655
Description: Lemma for ovolshft 21657. (Contributed by Mario Carneiro, 22-Mar-2014.)
Hypotheses
Ref Expression
ovolshft.1  |-  ( ph  ->  A  C_  RR )
ovolshft.2  |-  ( ph  ->  C  e.  RR )
ovolshft.3  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
ovolshft.4  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
ovolshft.5  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
ovolshft.6  |-  G  =  ( n  e.  NN  |->  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. )
ovolshft.7  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
ovolshft.8  |-  ( ph  ->  A  C_  U. ran  ( (,)  o.  F ) )
Assertion
Ref Expression
ovolshftlem1  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Distinct variable groups:    f, n, x, y, A    C, f, n, x, y    n, F, x    f, G, n, y    B, f, n, y    ph, f, n, y
Allowed substitution hints:    ph( x)    B( x)    S( x, y, f, n)    F( y, f)    G( x)    M( x, y, f, n)

Proof of Theorem ovolshftlem1
StepHypRef Expression
1 ovolshft.7 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2 ovolfcl 21613 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR  /\  ( 1st `  ( F `  n ) )  <_ 
( 2nd `  ( F `  n )
) ) )
31, 2sylan 471 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  e.  RR  /\  ( 2nd `  ( F `  n ) )  e.  RR  /\  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) ) )
43simp1d 1008 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  RR )
53simp2d 1009 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  RR )
6 ovolshft.2 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  RR )
76adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  RR )
83simp3d 1010 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) )
94, 5, 7, 8leadd1dd 10162 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  <_ 
( ( 2nd `  ( F `  n )
)  +  C ) )
10 df-br 4448 . . . . . . . . . . 11  |-  ( ( ( 1st `  ( F `  n )
)  +  C )  <_  ( ( 2nd `  ( F `  n
) )  +  C
)  <->  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.  e.  <_  )
119, 10sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  <_  )
124, 7readdcld 9619 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  e.  RR )
135, 7readdcld 9619 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( F `
 n ) )  +  C )  e.  RR )
14 opelxp 5028 . . . . . . . . . . 11  |-  ( <.
( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  ( RR 
X.  RR )  <->  ( (
( 1st `  ( F `  n )
)  +  C )  e.  RR  /\  (
( 2nd `  ( F `  n )
)  +  C )  e.  RR ) )
1512, 13, 14sylanbrc 664 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  ( RR 
X.  RR ) )
1611, 15elind 3688 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
17 ovolshft.6 . . . . . . . . 9  |-  G  =  ( n  e.  NN  |->  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. )
1816, 17fmptd 6043 . . . . . . . 8  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
19 eqid 2467 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
2019ovolfsf 21618 . . . . . . . 8  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  G ) : NN --> ( 0 [,) +oo ) )
21 ffn 5729 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  o.  G ) : NN --> ( 0 [,) +oo )  ->  ( ( abs  o.  -  )  o.  G )  Fn  NN )
2218, 20, 213syl 20 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  Fn  NN )
23 eqid 2467 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
2423ovolfsf 21618 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> ( 0 [,) +oo ) )
25 ffn 5729 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  o.  F ) : NN --> ( 0 [,) +oo )  ->  ( ( abs  o.  -  )  o.  F )  Fn  NN )
261, 24, 253syl 20 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  -  )  o.  F
)  Fn  NN )
27 opex 4711 . . . . . . . . . . . . . 14  |-  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  _V
2817fvmpt2 5955 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN  /\  <.
( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  _V )  ->  ( G `  n
)  =  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. )
2927, 28mpan2 671 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( G `  n )  =  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)
3029fveq2d 5868 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
31 ovex 6307 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) )  +  C )  e. 
_V
32 ovex 6307 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( F `
 n ) )  +  C )  e. 
_V
3331, 32op2nd 6790 . . . . . . . . . . . 12  |-  ( 2nd `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 2nd `  ( F `
 n ) )  +  C )
3430, 33syl6eq 2524 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
3529fveq2d 5868 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
3631, 32op1st 6789 . . . . . . . . . . . 12  |-  ( 1st `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 1st `  ( F `
 n ) )  +  C )
3735, 36syl6eq 2524 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
3834, 37oveq12d 6300 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( 2nd `  ( G `  n )
)  -  ( 1st `  ( G `  n
) ) )  =  ( ( ( 2nd `  ( F `  n
) )  +  C
)  -  ( ( 1st `  ( F `
 n ) )  +  C ) ) )
3938adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) ) )
405recnd 9618 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  CC )
414recnd 9618 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  CC )
427recnd 9618 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  CC )
4340, 41, 42pnpcan2d 9964 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4439, 43eqtrd 2508 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4519ovolfsval 21617 . . . . . . . . 9  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  n )  =  ( ( 2nd `  ( G `  n
) )  -  ( 1st `  ( G `  n ) ) ) )
4618, 45sylan 471 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( 2nd `  ( G `  n )
)  -  ( 1st `  ( G `  n
) ) ) )
4723ovolfsval 21617 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( ( abs  o.  -  )  o.  F
) `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
481, 47sylan 471 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  F ) `  n )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4944, 46, 483eqtr4d 2518 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( ( abs  o.  -  )  o.  F
) `  n )
)
5022, 26, 49eqfnfvd 5976 . . . . . 6  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  =  ( ( abs  o.  -  )  o.  F ) )
5150seqeq3d 12079 . . . . 5  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) ) )
52 ovolshft.5 . . . . 5  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
5351, 52syl6eqr 2526 . . . 4  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  =  S )
5453rneqd 5228 . . 3  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  =  ran  S )
5554supeq1d 7902 . 2  |-  ( ph  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  =  sup ( ran  S ,  RR* ,  <  ) )
56 ovolshft.3 . . . . . . . . 9  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
5756eleq2d 2537 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  <->  y  e.  { x  e.  RR  |  ( x  -  C )  e.  A } ) )
58 oveq1 6289 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  -  C )  =  ( y  -  C ) )
5958eleq1d 2536 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  -  C
)  e.  A  <->  ( y  -  C )  e.  A
) )
6059elrab 3261 . . . . . . . 8  |-  ( y  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )
6157, 60syl6bb 261 . . . . . . 7  |-  ( ph  ->  ( y  e.  B  <->  ( y  e.  RR  /\  ( y  -  C
)  e.  A ) ) )
6261biimpa 484 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )
63 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  -> 
( y  -  C
)  e.  A )
64 ovolshft.8 . . . . . . . . . 10  |-  ( ph  ->  A  C_  U. ran  ( (,)  o.  F ) )
65 ovolshft.1 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR )
66 ovolfioo 21614 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U. ran  ( (,)  o.  F )  <->  A. x  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) ) ) )
6765, 1, 66syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  U. ran  ( (,)  o.  F )  <->  A. x  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) ) ) )
6864, 67mpbid 210 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) ) )
6968adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  ->  A. x  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) ) )
70 breq2 4451 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
( 1st `  ( F `  n )
)  <  x  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
71 breq1 4450 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
x  <  ( 2nd `  ( F `  n
) )  <->  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) )
7270, 71anbi12d 710 . . . . . . . . . 10  |-  ( x  =  ( y  -  C )  ->  (
( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) )  <->  ( ( 1st `  ( F `  n ) )  < 
( y  -  C
)  /\  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) ) )
7372rexbidv 2973 . . . . . . . . 9  |-  ( x  =  ( y  -  C )  ->  ( E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) )  <->  E. n  e.  NN  ( ( 1st `  ( F `  n
) )  <  (
y  -  C )  /\  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) ) )
7473rspcv 3210 . . . . . . . 8  |-  ( ( y  -  C )  e.  A  ->  ( A. x  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) )  ->  E. n  e.  NN  ( ( 1st `  ( F `  n
) )  <  (
y  -  C )  /\  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) ) )
7563, 69, 74sylc 60 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  ->  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  ( y  -  C )  /\  (
y  -  C )  <  ( 2nd `  ( F `  n )
) ) )
7637adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
7776breq1d 4457 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( 1st `  ( G `  n )
)  <  y  <->  ( ( 1st `  ( F `  n ) )  +  C )  <  y
) )
784adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 1st `  ( F `  n ) )  e.  RR )
796ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  C  e.  RR )
80 simplrl 759 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  y  e.  RR )
8178, 79, 80ltaddsubd 10148 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( ( 1st `  ( F `  n )
)  +  C )  <  y  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
8277, 81bitrd 253 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( 1st `  ( G `  n )
)  <  y  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
8334adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
8483breq2d 4459 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
y  <  ( 2nd `  ( G `  n
) )  <->  y  <  ( ( 2nd `  ( F `  n )
)  +  C ) ) )
855adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n ) )  e.  RR )
8680, 79, 85ltsubaddd 10144 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( y  -  C
)  <  ( 2nd `  ( F `  n
) )  <->  y  <  ( ( 2nd `  ( F `  n )
)  +  C ) ) )
8784, 86bitr4d 256 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
y  <  ( 2nd `  ( G `  n
) )  <->  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) )
8882, 87anbi12d 710 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) )  <->  ( ( 1st `  ( F `  n ) )  < 
( y  -  C
)  /\  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) ) )
8988rexbidva 2970 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  -> 
( E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <  y  /\  y  <  ( 2nd `  ( G `  n
) ) )  <->  E. n  e.  NN  ( ( 1st `  ( F `  n
) )  <  (
y  -  C )  /\  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) ) )
9075, 89mpbird 232 . . . . . 6  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) )
9162, 90syldan 470 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <  y  /\  y  <  ( 2nd `  ( G `  n
) ) ) )
9291ralrimiva 2878 . . . 4  |-  ( ph  ->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) )
93 ssrab2 3585 . . . . . 6  |-  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR
9456, 93syl6eqss 3554 . . . . 5  |-  ( ph  ->  B  C_  RR )
95 ovolfioo 21614 . . . . 5  |-  ( ( B  C_  RR  /\  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( B  C_  U. ran  ( (,)  o.  G )  <->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) ) )
9694, 18, 95syl2anc 661 . . . 4  |-  ( ph  ->  ( B  C_  U. ran  ( (,)  o.  G )  <->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) ) )
9792, 96mpbird 232 . . 3  |-  ( ph  ->  B  C_  U. ran  ( (,)  o.  G ) )
98 ovolshft.4 . . . 4  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
99 eqid 2467 . . . 4  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
10098, 99elovolmr 21622 . . 3  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  B  C_ 
U. ran  ( (,)  o.  G ) )  ->  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) ,  RR* ,  <  )  e.  M )
10118, 97, 100syl2anc 661 . 2  |-  ( ph  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  M
)
10255, 101eqeltrrd 2556 1  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   <.cop 4033   U.cuni 4245   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ran crn 5000    o. ccom 5003    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780    ^m cmap 7417   supcsup 7896   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491   +oocpnf 9621   RR*cxr 9623    < clt 9624    <_ cle 9625    - cmin 9801   NNcn 10532   (,)cioo 11525   [,)cico 11527    seqcseq 12071   abscabs 13026
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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-iun 4327  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-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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-ioo 11529  df-ico 11531  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028
This theorem is referenced by:  ovolshftlem2  21656
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