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Theorem ovolshftlem1 21007
Description: Lemma for ovolshft 21009. (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 20965 . . . . . . . . . . . . . 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 1000 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  RR )
53simp2d 1001 . . . . . . . . . . . 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 1002 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) )
94, 5, 7, 8leadd1dd 9968 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  <_ 
( ( 2nd `  ( F `  n )
)  +  C ) )
10 df-br 4308 . . . . . . . . . . 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 9428 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  e.  RR )
135, 7readdcld 9428 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( F `
 n ) )  +  C )  e.  RR )
14 opelxp 4884 . . . . . . . . . . 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 3555 . . . . . . . . 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 5882 . . . . . . . 8  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
19 eqid 2443 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
2019ovolfsf 20970 . . . . . . . 8  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  G ) : NN --> ( 0 [,) +oo ) )
21 ffn 5574 . . . . . . . 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 2443 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
2423ovolfsf 20970 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> ( 0 [,) +oo ) )
25 ffn 5574 . . . . . . . 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 4571 . . . . . . . . . . . . . 14  |-  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  _V
2817fvmpt2 5796 . . . . . . . . . . . . . 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 5710 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
31 ovex 6131 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) )  +  C )  e. 
_V
32 ovex 6131 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( F `
 n ) )  +  C )  e. 
_V
3331, 32op2nd 6601 . . . . . . . . . . . 12  |-  ( 2nd `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 2nd `  ( F `
 n ) )  +  C )
3430, 33syl6eq 2491 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
3529fveq2d 5710 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
3631, 32op1st 6600 . . . . . . . . . . . 12  |-  ( 1st `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 1st `  ( F `
 n ) )  +  C )
3735, 36syl6eq 2491 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
3834, 37oveq12d 6124 . . . . . . . . . 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 9427 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  CC )
414recnd 9427 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  CC )
427recnd 9427 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  CC )
4340, 41, 42pnpcan2d 9772 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4439, 43eqtrd 2475 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4519ovolfsval 20969 . . . . . . . . 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 20969 . . . . . . . . 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 2485 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( ( abs  o.  -  )  o.  F
) `  n )
)
5022, 26, 49eqfnfvd 5815 . . . . . 6  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  =  ( ( abs  o.  -  )  o.  F ) )
5150seqeq3d 11829 . . . . 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 2493 . . . 4  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  =  S )
5453rneqd 5082 . . 3  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  =  ran  S )
5554supeq1d 7711 . 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 2510 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  <->  y  e.  { x  e.  RR  |  ( x  -  C )  e.  A } ) )
58 oveq1 6113 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  -  C )  =  ( y  -  C ) )
5958eleq1d 2509 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  -  C
)  e.  A  <->  ( y  -  C )  e.  A
) )
6059elrab 3132 . . . . . . . 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 20966 . . . . . . . . . . 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 4311 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
( 1st `  ( F `  n )
)  <  x  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
71 breq1 4310 . . . . . . . . . . 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 2751 . . . . . . . . 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 3084 . . . . . . . 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 4317 . . . . . . . . . 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 9954 . . . . . . . . . 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 4319 . . . . . . . . . 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 9950 . . . . . . . . . 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 2747 . . . . . . 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 2814 . . . 4  |-  ( ph  ->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) )
93 ssrab2 3452 . . . . . 6  |-  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR
9456, 93syl6eqss 3421 . . . . 5  |-  ( ph  ->  B  C_  RR )
95 ovolfioo 20966 . . . . 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 2443 . . . 4  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
10098, 99elovolmr 20974 . . 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 2518 1  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2730   E.wrex 2731   {crab 2734   _Vcvv 2987    i^i cin 3342    C_ wss 3343   <.cop 3898   U.cuni 4106   class class class wbr 4307    e. cmpt 4365    X. cxp 4853   ran crn 4856    o. ccom 4859    Fn wfn 5428   -->wf 5429   ` cfv 5433  (class class class)co 6106   1stc1st 6590   2ndc2nd 6591    ^m cmap 7229   supcsup 7705   RRcr 9296   0cc0 9297   1c1 9298    + caddc 9300   +oocpnf 9430   RR*cxr 9432    < clt 9433    <_ cle 9434    - cmin 9610   NNcn 10337   (,)cioo 11315   [,)cico 11317    seqcseq 11821   abscabs 12738
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-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-sup 7706  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-ioo 11319  df-ico 11321  df-fz 11453  df-seq 11822  df-exp 11881  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740
This theorem is referenced by:  ovolshftlem2  21008
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