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Theorem limsupval 11914
Description: The superior limit of an infinite sequence  F of extended real numbers, which is the infimum (indicated by  `'  <) of the set of suprema of all upper infinite subsequences of  F. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 5-Sep-2014.)
Hypothesis
Ref Expression
limsupval.1  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
Assertion
Ref Expression
limsupval  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
Distinct variable group:    k, F
Allowed substitution hints:    G( k)    V( k)

Proof of Theorem limsupval
StepHypRef Expression
1 elex 2771 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 imaeq1 4995 . . . . . . . . 9  |-  ( x  =  F  ->  (
x " ( k [,)  +oo ) )  =  ( F " (
k [,)  +oo ) ) )
32ineq1d 3344 . . . . . . . 8  |-  ( x  =  F  ->  (
( x " (
k [,)  +oo ) )  i^i  RR* )  =  ( ( F " (
k [,)  +oo ) )  i^i  RR* ) )
43supeq1d 7167 . . . . . . 7  |-  ( x  =  F  ->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  =  sup ( ( ( F " (
k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
54mpteq2dv 4081 . . . . . 6  |-  ( x  =  F  ->  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( k  e.  RR  |->  sup (
( ( F "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) )
6 limsupval.1 . . . . . 6  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
75, 6syl6eqr 2308 . . . . 5  |-  ( x  =  F  ->  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  G )
87rneqd 4894 . . . 4  |-  ( x  =  F  ->  ran  (  k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ran  G
)
98supeq1d 7167 . . 3  |-  ( x  =  F  ->  sup ( ran  (  k  e.  RR  |->  sup ( ( ( x " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
10 df-limsup 11911 . . 3  |-  limsup  =  ( x  e.  _V  |->  sup ( ran  (  k  e.  RR  |->  sup (
( ( x "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
11 xrltso 10443 . . . . 5  |-  <  Or  RR*
12 cnvso 5201 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 201 . . . 4  |-  `'  <  Or 
RR*
1413supex 7182 . . 3  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5536 . 2  |-  ( F  e.  _V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
161, 15syl 17 1  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   _Vcvv 2763    i^i cin 3126    e. cmpt 4051    Or wor 4285   `'ccnv 4660   ran crn 4662   "cima 4664   ` cfv 4673  (class class class)co 5792   supcsup 7161   RRcr 8704    +oocpnf 8832   RR*cxr 8834    < clt 8835   [,)cico 10625   limsupclsp 11910
This theorem is referenced by:  limsuple  11918  limsupval2  11920
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-pre-lttri 8779  ax-pre-lttrn 8780
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-limsup 11911
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