Theorem limsupval 15111

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 3440	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		imaeq1 5953	. . . . . . . . 9 ⊢ (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
3	2	ineq1d 4142	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
4	3	supeq1d 9135	. . . . . . 7 ⊢ (𝑥 = 𝐹 → sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
5	4	mpteq2dv 5172	. . . . . 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	eqtr4di 2797	. . . . 5 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
8	7	rneqd 5836	. . . 4 ⊢ (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
9	8	infeq1d 9166	. . 3 ⊢ (𝑥 = 𝐹 → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf(ran 𝐺, ℝ*, < ))
10		df-limsup 15108	. . 3 ⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
11		xrltso 12804	. . . 4 ⊢ < Or ℝ*
12	11	infex 9182	. . 3 ⊢ inf(ran 𝐺, ℝ*, < ) ∈ V
13	9, 10, 12	fvmpt 6857	. 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 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ↦ cmpt 5153  ran crn 5581   “ cima 5583  ‘cfv 6418  (class class class)co 7255  supcsup 9129  infcinf 9130  ℝcr 10801  +∞cpnf 10937  ℝ^*cxr 10939   < clt 10940  [,)cico 13010  lim supclsp 15107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-pre-lttri 10876  ax-pre-lttrn 10877

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-limsup 15108

This theorem is referenced by:  limsuple  15115  limsupval2  15117  limsupval3  43123  limsup0  43125  limsupresre  43127  limsuplesup  43130  limsuppnfdlem  43132  limsupres  43136  limsupvald  43186  limsupresxr  43197  liminfvalxr  43214