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Theorem ovolshftlem1 19406
Description: Lemma for ovolshft 19408. (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 19364 . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  e.  RR  /\  ( 2nd `  ( F `  n ) )  e.  RR  /\  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) ) )
43simp1d 970 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  RR )
53simp2d 971 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  RR )
6 ovolshft.2 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  RR )
76adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  RR )
83simp3d 972 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) )
94, 5, 7, 8leadd1dd 9641 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  <_ 
( ( 2nd `  ( F `  n )
)  +  C ) )
10 df-br 4214 . . . . . . . . . . 11  |-  ( ( ( 1st `  ( F `  n )
)  +  C )  <_  ( ( 2nd `  ( F `  n
) )  +  C
)  <->  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.  e.  <_  )
119, 10sylib 190 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  <_  )
124, 7readdcld 9116 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  e.  RR )
135, 7readdcld 9116 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( F `
 n ) )  +  C )  e.  RR )
14 opelxp 4909 . . . . . . . . . . 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 647 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  ( RR 
X.  RR ) )
16 elin 3531 . . . . . . . . . 10  |-  ( <.
( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <->  ( <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  <_  /\  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  ( RR 
X.  RR ) ) )
1711, 15, 16sylanbrc 647 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
18 ovolshft.6 . . . . . . . . 9  |-  G  =  ( n  e.  NN  |->  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. )
1917, 18fmptd 5894 . . . . . . . 8  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
20 eqid 2437 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
2120ovolfsf 19369 . . . . . . . 8  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  G ) : NN --> ( 0 [,)  +oo ) )
22 ffn 5592 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  o.  G ) : NN --> ( 0 [,) 
+oo )  ->  (
( abs  o.  -  )  o.  G )  Fn  NN )
2319, 21, 223syl 19 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  Fn  NN )
24 eqid 2437 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
2524ovolfsf 19369 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> ( 0 [,)  +oo ) )
26 ffn 5592 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  o.  F ) : NN --> ( 0 [,) 
+oo )  ->  (
( abs  o.  -  )  o.  F )  Fn  NN )
271, 25, 263syl 19 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  -  )  o.  F
)  Fn  NN )
28 opex 4428 . . . . . . . . . . . . . 14  |-  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  _V
2918fvmpt2 5813 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN  /\  <.
( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  _V )  ->  ( G `  n
)  =  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. )
3028, 29mpan2 654 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( G `  n )  =  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)
3130fveq2d 5733 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
32 ovex 6107 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) )  +  C )  e. 
_V
33 ovex 6107 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( F `
 n ) )  +  C )  e. 
_V
3432, 33op2nd 6357 . . . . . . . . . . . 12  |-  ( 2nd `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 2nd `  ( F `
 n ) )  +  C )
3531, 34syl6eq 2485 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
3630fveq2d 5733 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
3732, 33op1st 6356 . . . . . . . . . . . 12  |-  ( 1st `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 1st `  ( F `
 n ) )  +  C )
3836, 37syl6eq 2485 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
3935, 38oveq12d 6100 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( 2nd `  ( G `  n )
)  -  ( 1st `  ( G `  n
) ) )  =  ( ( ( 2nd `  ( F `  n
) )  +  C
)  -  ( ( 1st `  ( F `
 n ) )  +  C ) ) )
4039adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) ) )
415recnd 9115 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  CC )
424recnd 9115 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  CC )
437recnd 9115 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  CC )
4441, 42, 43pnpcan2d 9450 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4540, 44eqtrd 2469 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4620ovolfsval 19368 . . . . . . . . 9  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  n )  =  ( ( 2nd `  ( G `  n
) )  -  ( 1st `  ( G `  n ) ) ) )
4719, 46sylan 459 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( 2nd `  ( G `  n )
)  -  ( 1st `  ( G `  n
) ) ) )
4824ovolfsval 19368 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( ( abs  o.  -  )  o.  F
) `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
491, 48sylan 459 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  F ) `  n )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
5045, 47, 493eqtr4d 2479 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( ( abs  o.  -  )  o.  F
) `  n )
)
5123, 27, 50eqfnfvd 5831 . . . . . 6  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  =  ( ( abs  o.  -  )  o.  F ) )
5251seqeq3d 11332 . . . . 5  |-  ( ph  ->  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) ) )
53 ovolshft.5 . . . . 5  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
5452, 53syl6eqr 2487 . . . 4  |-  ( ph  ->  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  =  S )
5554rneqd 5098 . . 3  |-  ( ph  ->  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  =  ran  S )
5655supeq1d 7452 . 2  |-  ( ph  ->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  =  sup ( ran  S ,  RR* ,  <  ) )
57 ovolshft.3 . . . . . . . . 9  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
5857eleq2d 2504 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  <->  y  e.  { x  e.  RR  |  ( x  -  C )  e.  A } ) )
59 oveq1 6089 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  -  C )  =  ( y  -  C ) )
6059eleq1d 2503 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  -  C
)  e.  A  <->  ( y  -  C )  e.  A
) )
6160elrab 3093 . . . . . . . 8  |-  ( y  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )
6258, 61syl6bb 254 . . . . . . 7  |-  ( ph  ->  ( y  e.  B  <->  ( y  e.  RR  /\  ( y  -  C
)  e.  A ) ) )
6362biimpa 472 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )
64 simprr 735 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  -> 
( y  -  C
)  e.  A )
65 ovolshft.8 . . . . . . . . . 10  |-  ( ph  ->  A  C_  U. ran  ( (,)  o.  F ) )
66 ovolshft.1 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR )
67 ovolfioo 19365 . . . . . . . . . . 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 )
) ) ) )
6866, 1, 67syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  U. ran  ( (,)  o.  F )  <->  A. x  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) ) ) )
6965, 68mpbid 203 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) ) )
7069adantr 453 . . . . . . . 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 )
) ) )
71 breq2 4217 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
( 1st `  ( F `  n )
)  <  x  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
72 breq1 4216 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
x  <  ( 2nd `  ( F `  n
) )  <->  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) )
7371, 72anbi12d 693 . . . . . . . . . 10  |-  ( x  =  ( y  -  C )  ->  (
( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) )  <->  ( ( 1st `  ( F `  n ) )  < 
( y  -  C
)  /\  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) ) )
7473rexbidv 2727 . . . . . . . . 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 ) ) ) ) )
7574rspcv 3049 . . . . . . . 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 ) ) ) ) )
7664, 70, 75sylc 59 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  ->  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  ( y  -  C )  /\  (
y  -  C )  <  ( 2nd `  ( F `  n )
) ) )
7738adantl 454 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
7877breq1d 4223 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( 1st `  ( G `  n )
)  <  y  <->  ( ( 1st `  ( F `  n ) )  +  C )  <  y
) )
794adantlr 697 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 1st `  ( F `  n ) )  e.  RR )
806ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  C  e.  RR )
81 simplrl 738 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  y  e.  RR )
8279, 80, 81ltaddsubd 9627 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( ( 1st `  ( F `  n )
)  +  C )  <  y  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
8378, 82bitrd 246 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( 1st `  ( G `  n )
)  <  y  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
8435adantl 454 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
8584breq2d 4225 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
y  <  ( 2nd `  ( G `  n
) )  <->  y  <  ( ( 2nd `  ( F `  n )
)  +  C ) ) )
865adantlr 697 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n ) )  e.  RR )
8781, 80, 86ltsubaddd 9623 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( y  -  C
)  <  ( 2nd `  ( F `  n
) )  <->  y  <  ( ( 2nd `  ( F `  n )
)  +  C ) ) )
8885, 87bitr4d 249 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
y  <  ( 2nd `  ( G `  n
) )  <->  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) )
8983, 88anbi12d 693 . . . . . . . 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 ) ) ) ) )
9089rexbidva 2723 . . . . . . 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 ) ) ) ) )
9176, 90mpbird 225 . . . . . 6  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) )
9263, 91syldan 458 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <  y  /\  y  <  ( 2nd `  ( G `  n
) ) ) )
9392ralrimiva 2790 . . . 4  |-  ( ph  ->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) )
94 ssrab2 3429 . . . . . 6  |-  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR
9557, 94syl6eqss 3399 . . . . 5  |-  ( ph  ->  B  C_  RR )
96 ovolfioo 19365 . . . . 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 )
) ) ) )
9795, 19, 96syl2anc 644 . . . 4  |-  ( ph  ->  ( B  C_  U. ran  ( (,)  o.  G )  <->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) ) )
9893, 97mpbird 225 . . 3  |-  ( ph  ->  B  C_  U. ran  ( (,)  o.  G ) )
99 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* ,  <  ) ) }
100 eqid 2437 . . . 4  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
10199, 100elovolmr 19373 . . 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 )
10219, 98, 101syl2anc 644 . 2  |-  ( ph  ->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  M
)
10356, 102eqeltrrd 2512 1  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707   {crab 2710   _Vcvv 2957    i^i cin 3320    C_ wss 3321   <.cop 3818   U.cuni 4016   class class class wbr 4213    e. cmpt 4267    X. cxp 4877   ran crn 4880    o. ccom 4883    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082   1stc1st 6348   2ndc2nd 6349    ^m cmap 7019   supcsup 7446   RRcr 8990   0cc0 8991   1c1 8992    + caddc 8994    +oocpnf 9118   RR*cxr 9120    < clt 9121    <_ cle 9122    - cmin 9292   NNcn 10001   (,)cioo 10917   [,)cico 10919    seq cseq 11324   abscabs 12040
This theorem is referenced by:  ovolshftlem2  19407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-ioo 10921  df-ico 10923  df-fz 11045  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042
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