liminfvalxrmpt - Mathbox for Glauco Siliprandi

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Theorem liminfvalxrmpt 45757

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 5201	. . 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 45202	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ^*)
6	1, 2, 5	liminfvalxr 45754	. 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))))
7		eqidd 2730	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7, 4	fvmpt2d 6963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
9	8	xnegeqd 45406	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -_𝑒𝐵)
10	3, 9	mpteq2da 5194	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ -_𝑒𝐵))
11	10	fveq2d 6844	. . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
12	11	xnegeqd 45406	. 2 ⊢ (𝜑 → -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
13	6, 12	eqtrd 2764	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 1540 Ⅎwnf 1783 ∈ wcel 2109 ↦ cmpt 5183 ‘cfv 6499 ℝ^*cxr 11183 -_𝑒cxne 13045 lim supclsp 15412 lim infclsi 45722

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-xneg 13048 df-limsup 15413 df-liminf 45723

This theorem is referenced by: liminfval4 45760 liminfval3 45761

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