liminfval - Mathbox for Glauco Siliprandi

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Theorem liminfval 45741

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 45734	. 2 ⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
2		imaeq1 6010	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
3	2	ineq1d 4172	. . . . . . 7 ⊢ (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^))
4	3	infeq1d 9387	. . . . . 6 ⊢ (𝑥 = 𝐹 → inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
5	4	mpteq2dv 5189	. . . . 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 2767	. . . 4 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = 𝐺)
9	8	rneqd 5884	. . 3 ⊢ (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran 𝐺)
10	9	supeq1d 9355	. 2 ⊢ (𝑥 = 𝐹 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = sup(ran 𝐺, ℝ^, < ))
11		elex 3459	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
12		xrltso 13061	. . . 4 ⊢ < Or ℝ^*
13	12	supex 9373	. . 3 ⊢ sup(ran 𝐺, ℝ^*, < ) ∈ V
14	13	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑉 → sup(ran 𝐺, ℝ^*, < ) ∈ V)
15	1, 10, 11, 14	fvmptd3 6957	1 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3438 ∩ cin 3904 ↦ cmpt 5176 ran crn 5624 “ cima 5626 ‘cfv 6486 (class class class)co 7353 supcsup 9349 infcinf 9350 ℝcr 11027 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168 [,)cico 13268 lim infclsi 45733

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 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-pre-lttri 11102 ax-pre-lttrn 11103

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 3345 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-po 5531 df-so 5532 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-liminf 45734

This theorem is referenced by: liminfcl 45745 liminfvald 45746 liminfval5 45747 liminfresxr 45749 liminfval2 45750 liminfvalxr 45765

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