liminfgval - Mathbox for Glauco Siliprandi

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Theorem liminfgval 43338

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 7302	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝑘[,)+∞) = (𝑀[,)+∞))
2	1	imaeq2d 5970	. . . 4 ⊢ (𝑘 = 𝑀 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑀[,)+∞)))
3	2	ineq1d 4148	. . 3 ⊢ (𝑘 = 𝑀 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^))
4	3	infeq1d 9264	. 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 12903	. . 3 ⊢ < Or ℝ^*
7	6	infex 9280	. 2 ⊢ inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V
8	4, 5, 7	fvmpt 6895	1 ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 ∩ cin 3888 ↦ cmpt 5160 “ cima 5594 ‘cfv 6447 (class class class)co 7295 infcinf 9228 ℝcr 10898 +∞cpnf 11034 ℝ^*cxr 11036 < clt 11037 [,)cico 13109

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-pre-lttri 10973 ax-pre-lttrn 10974

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-sup 9229 df-inf 9230 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042

This theorem is referenced by: liminfval2 43344

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