liminfgval - Mathbox for Glauco Siliprandi

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Theorem liminfgval 46509

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 7423	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘[,)+∞) = (𝑀[,)+∞))
2	1	imaeq2d 6064	. . . 4 ⊢ (𝑘 = 𝑀 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑀[,)+∞)))
3	2	ineq1d 4172	. . 3 ⊢ (𝑘 = 𝑀 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^))
4	3	infeq1d 9441	. 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 13178	. . 3 ⊢ < Or ℝ^*
7	6	infex 9458	. 2 ⊢ inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V
8	4, 5, 7	fvmpt 6993	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 2146 ∩ cin 3905 ↦ cmpt 5194 “ cima 5666 ‘cfv 6540 (class class class)co 7416 infcinf 9404 ℝcr 11110 +∞cpnf 11251 ℝ^*cxr 11253 < clt 11254 [,)cico 13386

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-pre-lttri 11185 ax-pre-lttrn 11186

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7419 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9405 df-inf 9406 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259

This theorem is used by: liminfval2 46515

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