Theorem limsupval 15414

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 3493	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		imaeq1 6052	. . . . . . . . 9 ⊢ (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
3	2	ineq1d 4210	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
4	3	supeq1d 9437	. . . . . . 7 ⊢ (𝑥 = 𝐹 → sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
5	4	mpteq2dv 5249	. . . . . 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 2791	. . . . 5 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
8	7	rneqd 5935	. . . 4 ⊢ (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
9	8	infeq1d 9468	. . 3 ⊢ (𝑥 = 𝐹 → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = inf(ran 𝐺, ℝ*, < ))
10		df-limsup 15411	. . 3 ⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
11		xrltso 13116	. . . 4 ⊢ < Or ℝ*
12	11	infex 9484	. . 3 ⊢ inf(ran 𝐺, ℝ*, < ) ∈ V
13	9, 10, 12	fvmpt 6994	. 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 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3946   ↦ cmpt 5230  ran crn 5676   “ cima 5678  ‘cfv 6540  (class class class)co 7404  supcsup 9431  infcinf 9432  ℝcr 11105  +∞cpnf 11241  ℝ^*cxr 11243   < clt 11244  [,)cico 13322  lim supclsp 15410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-pre-lttri 11180  ax-pre-lttrn 11181

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-limsup 15411

This theorem is referenced by:  limsuple  15418  limsupval2  15420  limsupval3  44343  limsup0  44345  limsupresre  44347  limsuplesup  44350  limsuppnfdlem  44352  limsupres  44356  limsupvald  44406  limsupresxr  44417  liminfvalxr  44434