liminfcl - Mathbox for Glauco Siliprandi

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Theorem liminfcl 40789

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 2824	. . 3 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
2	1	liminfval 40785	. 2 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
3		nfv 2015	. . . 4 ⊢ Ⅎ𝑘 𝐹 ∈ 𝑉
4		inss2 4057	. . . . . 6 ⊢ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^
5		infxrcl 12450	. . . . . 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 40192	. . 3 ⊢ (𝐹 ∈ 𝑉 → ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*)
9	8	supxrcld 40104	. 2 ⊢ (𝐹 ∈ 𝑉 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^)
10	2, 9	eqeltrd 2905	1 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) ∈ ℝ^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 ∩ cin 3796 ⊆ wss 3797 ↦ cmpt 4951 ran crn 5342 “ cima 5344 ‘cfv 6122 (class class class)co 6904 supcsup 8614 infcinf 8615 ℝcr 10250 +∞cpnf 10387 ℝ^*cxr 10389 < clt 10390 [,)cico 12464 lim infclsi 40777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-sup 8616 df-inf 8617 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-liminf 40778

This theorem is referenced by: liminfcld 40796 climliminflimsupd 40827

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