liminfvalxrmpt - Mathbox for Glauco Siliprandi

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Theorem liminfvalxrmpt 42074

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 5166	. . 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 41512	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ^*)
6	1, 2, 5	liminfvalxr 42071	. 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))))
7		eqidd 2824	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7, 4	fvmpt2d 6783	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
9	8	xnegeqd 41718	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -_𝑒𝐵)
10	3, 9	mpteq2da 5162	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ -_𝑒𝐵))
11	10	fveq2d 6676	. . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
12	11	xnegeqd 41718	. 2 ⊢ (𝜑 → -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
13	6, 12	eqtrd 2858	1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ↦ cmpt 5148 ‘cfv 6357 ℝ^*cxr 10676 -_𝑒cxne 12507 lim supclsp 14829 lim infclsi 42039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-xneg 12510 df-limsup 14830 df-liminf 42040

This theorem is referenced by: liminfval4 42077 liminfval3 42078

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