liminfcl - Mathbox for Glauco Siliprandi

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Theorem liminfcl 43258

Description: Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Assertion**
Ref	Expression
liminfcl	⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) ∈ ℝ^*)

Proof of Theorem liminfcl Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2739	. . 3 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
2	1	liminfval 43254	. 2 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
3		nfv 1920	. . . 4 ⊢ Ⅎ𝑘 𝐹 ∈ 𝑉
4		inss2 4168	. . . . . 6 ⊢ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^
5		infxrcl 13049	. . . . . 6 ⊢ (((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
6	4, 5	ax-mp 5	. . . . 5 ⊢ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
7	6	a1i 11	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑘 ∈ ℝ) → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
8	3, 1, 7	rnmptssd 42688	. . 3 ⊢ (𝐹 ∈ 𝑉 → ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*)
9	8	supxrcld 42610	. 2 ⊢ (𝐹 ∈ 𝑉 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^)
10	2, 9	eqeltrd 2840	1 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) ∈ ℝ^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 ∩ cin 3890 ⊆ wss 3891 ↦ cmpt 5161 ran crn 5589 “ cima 5591 ‘cfv 6430 (class class class)co 7268 supcsup 9160 infcinf 9161 ℝcr 10854 +∞cpnf 10990 ℝ^*cxr 10992 < clt 10993 [,)cico 13063 lim infclsi 43246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-liminf 43247

This theorem is referenced by: liminfcld 43265 climliminflimsupd 43296

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