liminfvalxrmpt - Mathbox for Glauco Siliprandi

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Theorem liminfvalxrmpt 46214

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 5184	. . 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 45664	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ^*)
6	1, 2, 5	liminfvalxr 46211	. 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))))
7		eqidd 2737	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
8	7, 4	fvmpt2d 6961	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
9	8	xnegeqd 45865	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -_𝑒𝐵)
10	3, 9	mpteq2da 5177	. . . 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 45865	. 2 ⊢ (𝜑 → -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒𝐵)))
13	6, 12	eqtrd 2771	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 1542 Ⅎwnf 1785 ∈ wcel 2114 ↦ cmpt 5166 ‘cfv 6498 ℝ^*cxr 11178 -_𝑒cxne 13060 lim supclsp 15432 lim infclsi 46179

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 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 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-xneg 13063 df-limsup 15433 df-liminf 46180

This theorem is referenced by: liminfval4 46217 liminfval3 46218

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