Theorem limsupval 14665

Description: The superior limit of an infinite sequence 𝐹 of extended real numbers, which is the infimum 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 AV, 12-Sep-2014.)

Hypothesis

Ref	Expression
limsupval.1	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))

Assertion

Ref	Expression
limsupval	⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))

Distinct variable group:   𝑘,𝐹

Allowed substitution hints:   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem limsupval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3455	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		imaeq1 5801	. . . . . . . . 9 ⊢ (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
3	2	ineq1d 4108	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
4	3	supeq1d 8756	. . . . . . 7 ⊢ (𝑥 = 𝐹 → sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
5	4	mpteq2dv 5056	. . . . . 6 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
6		limsupval.1	. . . . . 6 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
7	5, 6	syl6eqr 2849	. . . . 5 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
8	7	rneqd 5690	. . . 4 ⊢ (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
9	8	infeq1d 8787	. . 3 ⊢ (𝑥 = 𝐹 → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf(ran 𝐺, ℝ*, < ))
10		df-limsup 14662	. . 3 ⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
11		xrltso 12384	. . . 4 ⊢ < Or ℝ*
12	11	infex 8803	. . 3 ⊢ inf(ran 𝐺, ℝ*, < ) ∈ V
13	9, 10, 12	fvmpt 6635	. 2 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
14	1, 13	syl 17	1 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))

Syntax hints:   → wi 4   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∩ cin 3858   ↦ cmpt 5041  ran crn 5444   “ cima 5446  ‘cfv 6225  (class class class)co 7016  supcsup 8750  infcinf 8751  ℝcr 10382  +∞cpnf 10518  ℝ^*cxr 10520   < clt 10521  [,)cico 12590  lim supclsp 14661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-pre-lttri 10457  ax-pre-lttrn 10458

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-po 5362  df-so 5363  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-sup 8752  df-inf 8753  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-limsup 14662

This theorem is referenced by:  limsuple  14669  limsupval2  14671  limsupval3  41515  limsup0  41517  limsupresre  41519  limsuplesup  41522  limsuppnfdlem  41524  limsupres  41528  limsupvald  41578  limsupresxr  41589  liminfvalxr  41606