liminfvalxrmpt - Mathbox for Glauco Siliprandi

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Theorem liminfvalxrmpt 45707

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 5274	. . 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 45142	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ^*)
6	1, 2, 5	liminfvalxr 45704	. 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))))
7		eqidd 2741	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7, 4	fvmpt2d 7042	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
9	8	xnegeqd 45352	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -_𝑒𝐵)
10	3, 9	mpteq2da 5264	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ -_𝑒𝐵))
11	10	fveq2d 6924	. . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
12	11	xnegeqd 45352	. 2 ⊢ (𝜑 → -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
13	6, 12	eqtrd 2780	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 1781 ∈ wcel 2108 ↦ cmpt 5249 ‘cfv 6573 ℝ^*cxr 11323 -_𝑒cxne 13172 lim supclsp 15516 lim infclsi 45672

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-xneg 13175 df-limsup 15517 df-liminf 45673

This theorem is referenced by: liminfval4 45710 liminfval3 45711

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