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Theorem limsupcl 15439

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 3468	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		df-limsup 15437	. . . 4 ⊢ lim sup = (𝑓 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
3		eqid 2729	. . . . . . 7 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
4		inss2 4201	. . . . . . . 8 ⊢ ((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^
5		supxrcl 13275	. . . . . . . 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 7084	. . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )):ℝ⟶ℝ^*
8		frn 6695	. . . . . 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 13294	. . . . 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 7084	. . 3 ⊢ lim sup:V⟶ℝ^*
13	12	ffvelcdmi 7055	. 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 2109 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ↦ cmpt 5188 ran crn 5639 “ cima 5641 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 supcsup 9391 infcinf 9392 ℝcr 11067 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 [,)cico 13308 lim supclsp 15436

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-limsup 15437

This theorem is referenced by: limsuplt 15445 limsupbnd1 15448 caucvgrlem 15639 limsupre 45639 limsupcld 45688 limsupcli 45755 limsupval4 45792 liminfreuzlem 45800

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