liminfgval - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfgval		Structured version Visualization version GIF version

Theorem liminfgval 45213

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 6058	. . . 4 ⊢ (𝑘 = 𝑀 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑀[,)+∞)))
3	2	ineq1d 4205	. . 3 ⊢ (𝑘 = 𝑀 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^))
4	3	infeq1d 9500	. 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 13152	. . 3 ⊢ < Or ℝ^*
7	6	infex 9516	. 2 ⊢ inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V
8	4, 5, 7	fvmpt 7000	1 ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ^), ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3938 ↦ cmpt 5226 “ cima 5675 ‘cfv 6543 (class class class)co 7416 infcinf 9464 ℝcr 11137 +∞cpnf 11275 ℝ^*cxr 11277 < clt 11278 [,)cico 13358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283

This theorem is referenced by: liminfval2 45219

W3C validator