MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolshftlem1 Unicode version

Theorem ovolshftlem1 18868
Description: Lemma for ovolshft 18870. (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 18826 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  e.  RR  /\  ( 2nd `  ( F `  n ) )  e.  RR  /\  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) ) )
43simp1d 967 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  RR )
53simp2d 968 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  RR )
6 ovolshft.2 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  RR )
76adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  RR )
83simp3d 969 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  <_  ( 2nd `  ( F `  n ) ) )
94, 5, 7, 8leadd1dd 9386 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  <_ 
( ( 2nd `  ( F `  n )
)  +  C ) )
10 df-br 4024 . . . . . . . . . . 11  |-  ( ( ( 1st `  ( F `  n )
)  +  C )  <_  ( ( 2nd `  ( F `  n
) )  +  C
)  <->  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.  e.  <_  )
119, 10sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  <_  )
124, 7readdcld 8862 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st `  ( F `
 n ) )  +  C )  e.  RR )
135, 7readdcld 8862 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( F `
 n ) )  +  C )  e.  RR )
14 opelxp 4719 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  ( RR 
X.  RR ) )
16 elin 3358 . . . . . . . . . 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 645 . . . . . . . . 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 5684 . . . . . . . 8  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
20 eqid 2283 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
2120ovolfsf 18831 . . . . . . . 8  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  G ) : NN --> ( 0 [,)  +oo ) )
22 ffn 5389 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  o.  G ) : NN --> ( 0 [,) 
+oo )  ->  (
( abs  o.  -  )  o.  G )  Fn  NN )
2319, 21, 223syl 18 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  Fn  NN )
24 eqid 2283 . . . . . . . . 9  |-  ( ( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  F )
2524ovolfsf 18831 . . . . . . . 8  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  F ) : NN --> ( 0 [,)  +oo ) )
26 ffn 5389 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  o.  F ) : NN --> ( 0 [,) 
+oo )  ->  (
( abs  o.  -  )  o.  F )  Fn  NN )
271, 25, 263syl 18 . . . . . . 7  |-  ( ph  ->  ( ( abs  o.  -  )  o.  F
)  Fn  NN )
28 opex 4237 . . . . . . . . . . . . . 14  |-  <. (
( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >.  e.  _V
2918fvmpt2 5608 . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  ( G `  n )  =  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)
3130fveq2d 5529 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
32 ovex 5883 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 n ) )  +  C )  e. 
_V
33 ovex 5883 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( F `
 n ) )  +  C )  e. 
_V
3432, 33op2nd 6129 . . . . . . . . . . . 12  |-  ( 2nd `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 2nd `  ( F `
 n ) )  +  C )
3531, 34syl6eq 2331 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
3630fveq2d 5529 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. ( ( 1st `  ( F `  n )
)  +  C ) ,  ( ( 2nd `  ( F `  n
) )  +  C
) >. ) )
3732, 33op1st 6128 . . . . . . . . . . . 12  |-  ( 1st `  <. ( ( 1st `  ( F `  n
) )  +  C
) ,  ( ( 2nd `  ( F `
 n ) )  +  C ) >.
)  =  ( ( 1st `  ( F `
 n ) )  +  C )
3836, 37syl6eq 2331 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
3935, 38oveq12d 5876 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( 2nd `  ( G `  n )
)  -  ( 1st `  ( G `  n
) ) )  =  ( ( ( 2nd `  ( F `  n
) )  +  C
)  -  ( ( 1st `  ( F `
 n ) )  +  C ) ) )
4039adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) ) )
415recnd 8861 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n
) )  e.  CC )
424recnd 8861 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( F `  n
) )  e.  CC )
437recnd 8861 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  C  e.  CC )
4441, 42, 43pnpcan2d 9195 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( 2nd `  ( F `  n )
)  +  C )  -  ( ( 1st `  ( F `  n
) )  +  C
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4540, 44eqtrd 2315 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  -  ( 1st `  ( G `  n )
) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4620ovolfsval 18830 . . . . . . . . 9  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  n )  =  ( ( 2nd `  ( G `  n
) )  -  ( 1st `  ( G `  n ) ) ) )
4719, 46sylan 457 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( 2nd `  ( G `  n )
)  -  ( 1st `  ( G `  n
) ) ) )
4824ovolfsval 18830 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( ( abs  o.  -  )  o.  F
) `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
491, 48sylan 457 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  F ) `  n )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
5045, 47, 493eqtr4d 2325 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  n )  =  ( ( ( abs  o.  -  )  o.  F
) `  n )
)
5123, 27, 50eqfnfvd 5625 . . . . . 6  |-  ( ph  ->  ( ( abs  o.  -  )  o.  G
)  =  ( ( abs  o.  -  )  o.  F ) )
5251seqeq3d 11054 . . . . 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 2333 . . . 4  |-  ( ph  ->  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  =  S )
5554rneqd 4906 . . 3  |-  ( ph  ->  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  =  ran  S )
5655supeq1d 7199 . 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 2350 . . . . . . . 8  |-  ( ph  ->  ( y  e.  B  <->  y  e.  { x  e.  RR  |  ( x  -  C )  e.  A } ) )
59 oveq1 5865 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  -  C )  =  ( y  -  C ) )
6059eleq1d 2349 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  -  C
)  e.  A  <->  ( y  -  C )  e.  A
) )
6160elrab 2923 . . . . . . . 8  |-  ( y  e.  { x  e.  RR  |  ( x  -  C )  e.  A }  <->  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )
6258, 61syl6bb 252 . . . . . . 7  |-  ( ph  ->  ( y  e.  B  <->  ( y  e.  RR  /\  ( y  -  C
)  e.  A ) ) )
6362biimpa 470 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )
64 simprr 733 . . . . . . . 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 18827 . . . . . . . . . . 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 642 . . . . . . . . . 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 201 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) ) )
7069adantr 451 . . . . . . . 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 4027 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
( 1st `  ( F `  n )
)  <  x  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
72 breq1 4026 . . . . . . . . . . 11  |-  ( x  =  ( y  -  C )  ->  (
x  <  ( 2nd `  ( F `  n
) )  <->  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) )
7371, 72anbi12d 691 . . . . . . . . . 10  |-  ( x  =  ( y  -  C )  ->  (
( ( 1st `  ( F `  n )
)  <  x  /\  x  <  ( 2nd `  ( F `  n )
) )  <->  ( ( 1st `  ( F `  n ) )  < 
( y  -  C
)  /\  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) ) )
7473rexbidv 2564 . . . . . . . . 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 2880 . . . . . . . 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 56 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  ->  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  ( y  -  C )  /\  (
y  -  C )  <  ( 2nd `  ( F `  n )
) ) )
7738adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 1st `  ( G `  n ) )  =  ( ( 1st `  ( F `  n )
)  +  C ) )
7877breq1d 4033 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( 1st `  ( G `  n )
)  <  y  <->  ( ( 1st `  ( F `  n ) )  +  C )  <  y
) )
794adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 1st `  ( F `  n ) )  e.  RR )
806ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  C  e.  RR )
81 simplrl 736 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  y  e.  RR )
8279, 80, 81ltaddsubd 9372 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( ( 1st `  ( F `  n )
)  +  C )  <  y  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
8378, 82bitrd 244 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( 1st `  ( G `  n )
)  <  y  <->  ( 1st `  ( F `  n
) )  <  (
y  -  C ) ) )
8435adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n ) )  =  ( ( 2nd `  ( F `  n )
)  +  C ) )
8584breq2d 4035 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
y  <  ( 2nd `  ( G `  n
) )  <->  y  <  ( ( 2nd `  ( F `  n )
)  +  C ) ) )
865adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  ( 2nd `  ( F `  n ) )  e.  RR )
8781, 80, 86ltsubaddd 9368 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
( y  -  C
)  <  ( 2nd `  ( F `  n
) )  <->  y  <  ( ( 2nd `  ( F `  n )
)  +  C ) ) )
8885, 87bitr4d 247 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  RR  /\  ( y  -  C
)  e.  A ) )  /\  n  e.  NN )  ->  (
y  <  ( 2nd `  ( G `  n
) )  <->  ( y  -  C )  <  ( 2nd `  ( F `  n ) ) ) )
8983, 88anbi12d 691 . . . . . . . 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 2560 . . . . . . 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 223 . . . . . 6  |-  ( (
ph  /\  ( y  e.  RR  /\  ( y  -  C )  e.  A ) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) )
9263, 91syldan 456 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <  y  /\  y  <  ( 2nd `  ( G `  n
) ) ) )
9392ralrimiva 2626 . . . 4  |-  ( ph  ->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) )
94 ssrab2 3258 . . . . . . 7  |-  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR
9594a1i 10 . . . . . 6  |-  ( ph  ->  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR )
9657, 95eqsstrd 3212 . . . . 5  |-  ( ph  ->  B  C_  RR )
97 ovolfioo 18827 . . . . 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 )
) ) ) )
9896, 19, 97syl2anc 642 . . . 4  |-  ( ph  ->  ( B  C_  U. ran  ( (,)  o.  G )  <->  A. y  e.  B  E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <  y  /\  y  <  ( 2nd `  ( G `  n )
) ) ) )
9993, 98mpbird 223 . . 3  |-  ( ph  ->  B  C_  U. ran  ( (,)  o.  G ) )
100 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* ,  <  ) ) }
101 eqid 2283 . . . 4  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
102100, 101elovolmr 18835 . . 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 )
10319, 99, 102syl2anc 642 . 2  |-  ( ph  ->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  M
)
10456, 103eqeltrrd 2358 1  |-  ( ph  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   <.cop 3643   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   (,)cioo 10656   [,)cico 10658    seq cseq 11046   abscabs 11719
This theorem is referenced by:  ovolshftlem2  18869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioo 10660  df-ico 10662  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
  Copyright terms: Public domain W3C validator