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Theorem ovolshftlem1 22212
Description: Lemma for ovolshft 22214. (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 22170 . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  e.  RR  /\  ( 2nd `  ( F `  n ) )  e.  RR  /\  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) ) )
43simp1d 1009 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  RR )
53simp2d 1010 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  RR )
6 ovolshft.2 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  RR )
76adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  RR )
83simp3d 1011 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) )
94, 5, 7, 8leadd1dd 10206 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  <_ 
( ( 2nd `  ( F `  n )
)  +  C ) )
10 df-br 4396 . . . . . . . . . . 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 9653 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  e.  RR )
135, 7readdcld 9653 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( F `
 n ) )  +  C )  e.  RR )
14 opelxp 4853 . . . . . . . . . . 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 662 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  ( RR 
X.  RR ) )
1611, 15elind 3627 . . . . . . . . 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 6033 . . . . . . . 8  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
19 eqid 2402 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
2019ovolfsf 22175 . . . . . . . 8  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  G ) : NN --> ( 0 [,) +oo ) )
21 ffn 5714 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  o.  G ) : NN --> ( 0 [,) +oo )  ->  ( ( abs  o.  -  )  o.  G )  Fn  NN )
2218, 20, 213syl 18 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  Fn  NN )
23 eqid 2402 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
2423ovolfsf 22175 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> ( 0 [,) +oo ) )
25 ffn 5714 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  o.  F ) : NN --> ( 0 [,) +oo )  ->  ( ( abs  o.  -  )  o.  F )  Fn  NN )
261, 24, 253syl 18 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  -  )  o.  F
)  Fn  NN )
27 opex 4655 . . . . . . . . . . . . . 14  |-  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  _V
2817fvmpt2 5941 . . . . . . . . . . . . . 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 669 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( G `  n )  =  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)
3029fveq2d 5853 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
31 ovex 6306 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) )  +  C )  e. 
_V
32 ovex 6306 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( F `
 n ) )  +  C )  e. 
_V
3331, 32op2nd 6793 . . . . . . . . . . . 12  |-  ( 2nd `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 2nd `  ( F `
 n ) )  +  C )
3430, 33syl6eq 2459 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
3529fveq2d 5853 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
3631, 32op1st 6792 . . . . . . . . . . . 12  |-  ( 1st `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 1st `  ( F `
 n ) )  +  C )
3735, 36syl6eq 2459 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
3834, 37oveq12d 6296 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( 2nd `  ( G `  n )
)  -  ( 1st `  ( G `  n
) ) )  =  ( ( ( 2nd `  ( F `  n
) )  +  C
)  -  ( ( 1st `  ( F `
 n ) )  +  C ) ) )
3938adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) ) )
405recnd 9652 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  CC )
414recnd 9652 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  CC )
427recnd 9652 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  CC )
4340, 41, 42pnpcan2d 10005 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4439, 43eqtrd 2443 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4519ovolfsval 22174 . . . . . . . . 9  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  n )  =  ( ( 2nd `  ( G `  n
) )  -  ( 1st `  ( G `  n ) ) ) )
4618, 45sylan 469 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( 2nd `  ( G `  n )
)  -  ( 1st `  ( G `  n
) ) ) )
4723ovolfsval 22174 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( ( abs  o.  -  )  o.  F
) `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
481, 47sylan 469 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  F ) `  n )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4944, 46, 483eqtr4d 2453 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( ( abs  o.  -  )  o.  F
) `  n )
)
5022, 26, 49eqfnfvd 5962 . . . . . 6  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  =  ( ( abs  o.  -  )  o.  F ) )
5150seqeq3d 12159 . . . . 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 2461 . . . 4  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  =  S )
5453rneqd 5051 . . 3  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  =  ran  S )
5554supeq1d 7939 . 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 2472 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  <->  y  e.  { x  e.  RR  |  ( x  -  C )  e.  A } ) )
58 oveq1 6285 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  -  C )  =  ( y  -  C ) )
5958eleq1d 2471 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  -  C
)  e.  A  <->  ( y  -  C )  e.  A
) )
6059elrab 3207 . . . . . . . 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 482 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )
63 simprr 758 . . . . . . . 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 22171 . . . . . . . . . . 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 659 . . . . . . . . . 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 463 . . . . . . . 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 4399 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
( 1st `  ( F `  n )
)  <  x  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
71 breq1 4398 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
x  <  ( 2nd `  ( F `  n
) )  <->  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) )
7270, 71anbi12d 709 . . . . . . . . . 10  |-  ( x  =  ( y  -  C )  ->  (
( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) )  <->  ( ( 1st `  ( F `  n ) )  < 
( y  -  C
)  /\  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) ) )
7372rexbidv 2918 . . . . . . . . 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 3156 . . . . . . . 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 59 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  ->  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  ( y  -  C )  /\  (
y  -  C )  <  ( 2nd `  ( F `  n )
) ) )
7637adantl 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
7776breq1d 4405 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( 1st `  ( G `  n )
)  <  y  <->  ( ( 1st `  ( F `  n ) )  +  C )  <  y
) )
784adantlr 713 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 1st `  ( F `  n ) )  e.  RR )
796ad2antrr 724 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  C  e.  RR )
80 simplrl 762 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  y  e.  RR )
8178, 79, 80ltaddsubd 10192 . . . . . . . . . 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 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
8483breq2d 4407 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
y  <  ( 2nd `  ( G `  n
) )  <->  y  <  ( ( 2nd `  ( F `  n )
)  +  C ) ) )
855adantlr 713 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n ) )  e.  RR )
8680, 79, 85ltsubaddd 10188 . . . . . . . . . 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 709 . . . . . . . 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 2915 . . . . . . 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 468 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <  y  /\  y  <  ( 2nd `  ( G `  n
) ) ) )
9291ralrimiva 2818 . . . 4  |-  ( ph  ->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) )
93 ssrab2 3524 . . . . . 6  |-  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR
9456, 93syl6eqss 3492 . . . . 5  |-  ( ph  ->  B  C_  RR )
95 ovolfioo 22171 . . . . 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 659 . . . 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 2402 . . . 4  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
10098, 99elovolmr 22179 . . 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 659 . 2  |-  ( ph  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  M
)
10255, 101eqeltrrd 2491 1  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   {crab 2758   _Vcvv 3059    i^i cin 3413    C_ wss 3414   <.cop 3978   U.cuni 4191   class class class wbr 4395    |-> cmpt 4453    X. cxp 4821   ran crn 4824    o. ccom 4827    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783    ^m cmap 7457   supcsup 7934   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525   +oocpnf 9655   RR*cxr 9657    < clt 9658    <_ cle 9659    - cmin 9841   NNcn 10576   (,)cioo 11582   [,)cico 11584    seqcseq 12151   abscabs 13216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-ioo 11586  df-ico 11588  df-fz 11727  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218
This theorem is referenced by:  ovolshftlem2  22213
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