liminfvalxrmpt - Mathbox for Glauco Siliprandi

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

Theorem liminfvalxrmpt 45742

Description: Alternate definition of lim inf when 𝐹 is an extended real-valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
liminfvalxrmpt.1	⊢ Ⅎ𝑥𝜑
liminfvalxrmpt.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
liminfvalxrmpt.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)

**Assertion**
Ref	Expression
liminfvalxrmpt	⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))

Distinct variable group: 𝑥,𝐴 Allowed substitution hints: 𝜑(𝑥) 𝐵(𝑥) 𝑉(𝑥)

**Proof of Theorem liminfvalxrmpt**
Step	Hyp	Ref	Expression
1		nfmpt1 5256	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
2		liminfvalxrmpt.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		liminfvalxrmpt.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		liminfvalxrmpt.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
5	3, 4	fmptd2f 45178	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ^*)
6	1, 2, 5	liminfvalxr 45739	. 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))))
7		eqidd 2736	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7, 4	fvmpt2d 7029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
9	8	xnegeqd 45387	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -_𝑒𝐵)
10	3, 9	mpteq2da 5246	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ -_𝑒𝐵))
11	10	fveq2d 6911	. . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
12	11	xnegeqd 45387	. 2 ⊢ (𝜑 → -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
13	6, 12	eqtrd 2775	1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 ℝ^*cxr 11292 -_𝑒cxne 13149 lim supclsp 15503 lim infclsi 45707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-xneg 13152 df-limsup 15504 df-liminf 45708

This theorem is referenced by: liminfval4 45745 liminfval3 45746

W3C validator