liminfgval - Mathbox for Glauco Siliprandi

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Theorem liminfgval 46590

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

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

**Assertion**
Ref	Expression
liminfgval	⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ))

Distinct variable groups: 𝑘,𝐹 𝑘,𝑀 Allowed substitution hint: 𝐺(𝑘)

**Proof of Theorem liminfgval**
Step	Hyp	Ref	Expression
1		oveq1 7420	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘[,)+∞) = (𝑀[,)+∞))
2	1	imaeq2d 6056	. . . 4 ⊢ (𝑘 = 𝑀 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑀[,)+∞)))
3	2	ineq1d 4165	. . 3 ⊢ (𝑘 = 𝑀 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^))
4	3	infeq1d 9448	. 2 ⊢ (𝑘 = 𝑀 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ))
5		liminfgval.1	. 2 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
6		xrltso 13192	. . 3 ⊢ < Or ℝ^*
7	6	infex 9465	. 2 ⊢ inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V
8	4, 5, 7	fvmpt 6986	1 ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ∩ cin 3898 ↦ cmpt 5186 “ cima 5658 ‘cfv 6533 (class class class)co 7413 infcinf 9411 ℝcr 11123 +∞cpnf 11264 ℝ^*cxr 11266 < clt 11267 [,)cico 13400

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-pre-lttri 11198 ax-pre-lttrn 11199

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7416 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272

This theorem is used by: liminfval2 46596

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