liminfresico - Mathbox for Glauco Siliprandi

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Theorem liminfresico 45082

Description: The inferior limit doesn't change when a function is restricted to an upperset of reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
liminfresico.1	⊢ (𝜑 → 𝑀 ∈ ℝ)
liminfresico.2	⊢ 𝑍 = (𝑀[,)+∞)
liminfresico.3	⊢ (𝜑 → 𝐹 ∈ 𝑉)

**Assertion**
Ref	Expression
liminfresico	⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹))

Proof of Theorem liminfresico Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		liminfresico.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
2	1	rexrd 11286	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
3	2	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑀 ∈ ℝ^*)
4		pnfxr 11290	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
5	4	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → +∞ ∈ ℝ^*)
6		ressxr 11280	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
7	4	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → +∞ ∈ ℝ^*)
8		icossre 13429	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℝ ∧ +∞ ∈ ℝ^*) → (𝑀[,)+∞) ⊆ ℝ)
9	1, 7, 8	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀[,)+∞) ⊆ ℝ)
10	9	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀[,)+∞) ⊆ ℝ)
11		liminfresico.2	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑍 = (𝑀[,)+∞)
12	11	eleq2i 2820	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (𝑀[,)+∞))
13	12	biimpi 215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ (𝑀[,)+∞))
14	13	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (𝑀[,)+∞))
15	10, 14	sseldd 3979	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℝ)
16	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑘 ∈ ℝ)
17		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ (𝑘[,)+∞))
18		elicore 13400	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ ℝ)
19	16, 17, 18	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ ℝ)
20	6, 19	sselid 3976	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ ℝ^*)
21	1	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑀 ∈ ℝ)
22	2	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ∈ ℝ^*)
23	4	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → +∞ ∈ ℝ^*)
24	22, 23, 14	icogelbd 44866	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ≤ 𝑘)
25	24	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑀 ≤ 𝑘)
26	6, 16	sselid 3976	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑘 ∈ ℝ^*)
27	26, 5, 17	icogelbd 44866	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑘 ≤ 𝑦)
28	21, 16, 19, 25, 27	letrd 11393	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑀 ≤ 𝑦)
29	19	ltpnfd 13125	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 < +∞)
30	3, 5, 20, 28, 29	elicod 13398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ (𝑀[,)+∞))
31	30, 11	eleqtrrdi 2839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ 𝑍)
32	31	ssd 44369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘[,)+∞) ⊆ 𝑍)
33		resima2 6014	. . . . . . . . 9 ⊢ ((𝑘[,)+∞) ⊆ 𝑍 → ((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
34	32, 33	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
35	34	ineq1d 4207	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^))
36	35	infeq1d 9492	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
37	36	mpteq2dva 5242	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ 𝑍 ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
38	37	rneqd 5934	. . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝑍 ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran (𝑘 ∈ 𝑍 ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
39	11, 9	eqsstrid 4026	. . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
40	39	mptimass 6070	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍) = ran (𝑘 ∈ 𝑍 ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
41	39	mptimass 6070	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍) = ran (𝑘 ∈ 𝑍 ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
42	38, 40, 41	3eqtr4d 2777	. . 3 ⊢ (𝜑 → ((𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍) = ((𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍))
43	42	supeq1d 9461	. 2 ⊢ (𝜑 → sup(((𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍), ℝ^, < ) = sup(((𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍), ℝ^, < ))
44		eqid 2727	. . 3 ⊢ (𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
45		liminfresico.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
46	45	resexd 6026	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑍) ∈ V)
47	11	supeq1i 9462	. . . . 5 ⊢ sup(𝑍, ℝ^, < ) = sup((𝑀[,)+∞), ℝ^, < )
48	47	a1i 11	. . . 4 ⊢ (𝜑 → sup(𝑍, ℝ^, < ) = sup((𝑀[,)+∞), ℝ^, < ))
49	1	renepnfd 11287	. . . . 5 ⊢ (𝜑 → 𝑀 ≠ +∞)
50		icopnfsup 13854	. . . . 5 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≠ +∞) → sup((𝑀[,)+∞), ℝ^*, < ) = +∞)
51	2, 49, 50	syl2anc 583	. . . 4 ⊢ (𝜑 → sup((𝑀[,)+∞), ℝ^*, < ) = +∞)
52	48, 51	eqtrd 2767	. . 3 ⊢ (𝜑 → sup(𝑍, ℝ^*, < ) = +∞)
53	44, 46, 39, 52	liminfval2 45079	. 2 ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = sup(((𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍), ℝ^*, < ))
54		eqid 2727	. . 3 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
55	54, 45, 39, 52	liminfval2 45079	. 2 ⊢ (𝜑 → (lim inf‘𝐹) = sup(((𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍), ℝ^*, < ))
56	43, 53, 55	3eqtr4d 2777	1 ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 Vcvv 3469 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5142 ↦ cmpt 5225 ran crn 5673 ↾ cres 5674 “ cima 5675 ‘cfv 6542 (class class class)co 7414 supcsup 9455 infcinf 9456 ℝcr 11129 +∞cpnf 11267 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 [,)cico 13350 lim infclsi 45062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-q 12955 df-ico 13354 df-liminf 45063

This theorem is referenced by: liminfresre 45090 liminfresicompt 45091 liminfresuz 45095

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