liminfval - Mathbox for Glauco Siliprandi

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Theorem liminfval 45788

Description: The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypothesis**
Ref	Expression
liminfval.1	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))

**Assertion**
Ref	Expression
liminfval	⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ^*, < ))

Distinct variable group: 𝑘,𝐹 Allowed substitution hints: 𝐺(𝑘) 𝑉(𝑘)

Proof of Theorem liminfval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-liminf 45781	. 2 ⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
2		imaeq1 6042	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
3	2	ineq1d 4194	. . . . . . 7 ⊢ (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^))
4	3	infeq1d 9490	. . . . . 6 ⊢ (𝑥 = 𝐹 → inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
5	4	mpteq2dv 5215	. . . . 5 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
6		liminfval.1	. . . . . 6 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
7	6	a1i 11	. . . . 5 ⊢ (𝑥 = 𝐹 → 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
8	5, 7	eqtr4d 2773	. . . 4 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = 𝐺)
9	8	rneqd 5918	. . 3 ⊢ (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran 𝐺)
10	9	supeq1d 9458	. 2 ⊢ (𝑥 = 𝐹 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = sup(ran 𝐺, ℝ^, < ))
11		elex 3480	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
12		xrltso 13157	. . . 4 ⊢ < Or ℝ^*
13	12	supex 9476	. . 3 ⊢ sup(ran 𝐺, ℝ^*, < ) ∈ V
14	13	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑉 → sup(ran 𝐺, ℝ^*, < ) ∈ V)
15	1, 10, 11, 14	fvmptd3 7009	1 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3459 ∩ cin 3925 ↦ cmpt 5201 ran crn 5655 “ cima 5657 ‘cfv 6531 (class class class)co 7405 supcsup 9452 infcinf 9453 ℝcr 11128 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 [,)cico 13364 lim infclsi 45780

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 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-pre-lttri 11203 ax-pre-lttrn 11204

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-liminf 45781

This theorem is referenced by: liminfcl 45792 liminfvald 45793 liminfval5 45794 liminfresxr 45796 liminfval2 45797 liminfvalxr 45812

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