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Definition df-limsup 13051
Description: Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 13054 for its value. (Contributed by NM, 26-Oct-2005.)
Assertion
Ref Expression
df-limsup  |-  limsup  =  ( x  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( x "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
Distinct variable group:    x, k

Detailed syntax breakdown of Definition df-limsup
StepHypRef Expression
1 clsp 13050 . 2  class  limsup
2 vx . . 3  setvar  x
3 cvv 3068 . . 3  class  _V
4 vk . . . . . 6  setvar  k
5 cr 9382 . . . . . 6  class  RR
62cv 1369 . . . . . . . . 9  class  x
74cv 1369 . . . . . . . . . 10  class  k
8 cpnf 9516 . . . . . . . . . 10  class +oo
9 cico 11403 . . . . . . . . . 10  class  [,)
107, 8, 9co 6190 . . . . . . . . 9  class  ( k [,) +oo )
116, 10cima 4941 . . . . . . . 8  class  ( x
" ( k [,) +oo ) )
12 cxr 9518 . . . . . . . 8  class  RR*
1311, 12cin 3425 . . . . . . 7  class  ( ( x " ( k [,) +oo ) )  i^i  RR* )
14 clt 9519 . . . . . . 7  class  <
1513, 12, 14csup 7791 . . . . . 6  class  sup (
( ( x "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )
164, 5, 15cmpt 4448 . . . . 5  class  ( k  e.  RR  |->  sup (
( ( x "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
1716crn 4939 . . . 4  class  ran  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
1814ccnv 4937 . . . 4  class  `'  <
1917, 12, 18csup 7791 . . 3  class  sup ( ran  ( k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )
202, 3, 19cmpt 4448 . 2  class  ( x  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
211, 20wceq 1370 1  wff  limsup  =  ( x  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( x "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
Colors of variables: wff setvar class
This definition is referenced by:  limsupcl  13053  limsupval  13054
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