limsupcl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > limsupcl		Structured version Visualization version GIF version

Theorem limsupcl 14403

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 3352	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		df-limsup 14401	. . . 4 ⊢ lim sup = (𝑓 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
3		eqid 2760	. . . . . . 7 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
4		inss2 3977	. . . . . . . 8 ⊢ ((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^
5		supxrcl 12338	. . . . . . . 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 6546	. . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )):ℝ⟶ℝ^*
8		frn 6214	. . . . . 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 12356	. . . . 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 6546	. . 3 ⊢ lim sup:V⟶ℝ^*
13	12	ffvelrni 6521	. 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 2139 Vcvv 3340 ∩ cin 3714 ⊆ wss 3715 ↦ cmpt 4881 ran crn 5267 “ cima 5269 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 supcsup 8511 infcinf 8512 ℝcr 10127 +∞cpnf 10263 ℝ^*cxr 10265 < clt 10266 [,)cico 12370 lim supclsp 14400

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-limsup 14401

This theorem is referenced by: limsuplt 14409 limsupbnd1 14412 caucvgrlem 14602 limsupre 40376 limsupcld 40425 limsupcli 40492 limsupval4 40529 liminfreuzlem 40537

W3C validator