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Theorem limsupcl 15509

Description: Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2020.)

**Assertion**
Ref	Expression
limsupcl	⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) ∈ ℝ^*)

Proof of Theorem limsupcl Dummy variables 𝑘 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3501	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		df-limsup 15507	. . . 4 ⊢ lim sup = (𝑓 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
3		eqid 2737	. . . . . . 7 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
4		inss2 4238	. . . . . . . 8 ⊢ ((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^
5		supxrcl 13357	. . . . . . . 8 ⊢ (((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
6	4, 5	mp1i 13	. . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
7	3, 6	fmpti 7132	. . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )):ℝ⟶ℝ^*
8		frn 6743	. . . . . 6 ⊢ ((𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )):ℝ⟶ℝ^* → ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*)
9	7, 8	ax-mp 5	. . . . 5 ⊢ ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*
10		infxrcl 13375	. . . . 5 ⊢ (ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^* → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^)
11	9, 10	mp1i 13	. . . 4 ⊢ (𝑓 ∈ V → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^)
12	2, 11	fmpti 7132	. . 3 ⊢ lim sup:V⟶ℝ^*
13	12	ffvelcdmi 7103	. 2 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ^*)
14	1, 13	syl 17	1 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) ∈ ℝ^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ↦ cmpt 5225 ran crn 5686 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 infcinf 9481 ℝcr 11154 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 [,)cico 13389 lim supclsp 15506

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-limsup 15507

This theorem is referenced by: limsuplt 15515 limsupbnd1 15518 caucvgrlem 15709 limsupre 45656 limsupcld 45705 limsupcli 45772 limsupval4 45809 liminfreuzlem 45817

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