liminfresico - Mathbox for Glauco Siliprandi

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Theorem liminfresico 44473

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 11260	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
3	2	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑀 ∈ ℝ^*)
4		pnfxr 11264	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
5	4	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → +∞ ∈ ℝ^*)
6		ressxr 11254	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
7	4	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → +∞ ∈ ℝ^*)
8		icossre 13401	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℝ ∧ +∞ ∈ ℝ^*) → (𝑀[,)+∞) ⊆ ℝ)
9	1, 7, 8	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀[,)+∞) ⊆ ℝ)
10	9	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀[,)+∞) ⊆ ℝ)
11		liminfresico.2	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑍 = (𝑀[,)+∞)
12	11	eleq2i 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (𝑀[,)+∞))
13	12	biimpi 215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ (𝑀[,)+∞))
14	13	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (𝑀[,)+∞))
15	10, 14	sseldd 3982	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℝ)
16	15	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑘 ∈ ℝ)
17		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ (𝑘[,)+∞))
18		elicore 13372	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ ℝ)
19	16, 17, 18	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ ℝ)
20	6, 19	sselid 3979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ ℝ^*)
21	1	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑀 ∈ ℝ)
22	2	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ∈ ℝ^*)
23	4	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → +∞ ∈ ℝ^*)
24	22, 23, 14	icogelbd 44257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ≤ 𝑘)
25	24	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑀 ≤ 𝑘)
26	6, 16	sselid 3979	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑘 ∈ ℝ^*)
27	26, 5, 17	icogelbd 44257	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑘 ≤ 𝑦)
28	21, 16, 19, 25, 27	letrd 11367	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑀 ≤ 𝑦)
29	19	ltpnfd 13097	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 < +∞)
30	3, 5, 20, 28, 29	elicod 13370	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ (𝑀[,)+∞))
31	30, 11	eleqtrrdi 2844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑦 ∈ (𝑘[,)+∞)) → 𝑦 ∈ 𝑍)
32	31	ssd 43754	. . . . . . . . 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 4210	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^))
36	35	infeq1d 9468	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
37	36	mpteq2dva 5247	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ 𝑍 ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
38	37	rneqd 5935	. . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝑍 ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran (𝑘 ∈ 𝑍 ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
39	11, 9	eqsstrid 4029	. . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℝ)
40	39	mptima2 43935	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍) = ran (𝑘 ∈ 𝑍 ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
41	39	mptima2 43935	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍) = ran (𝑘 ∈ 𝑍 ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
42	38, 40, 41	3eqtr4d 2782	. . 3 ⊢ (𝜑 → ((𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍) = ((𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍))
43	42	supeq1d 9437	. 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 2732	. . 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 9438	. . . . 5 ⊢ sup(𝑍, ℝ^, < ) = sup((𝑀[,)+∞), ℝ^, < )
48	47	a1i 11	. . . 4 ⊢ (𝜑 → sup(𝑍, ℝ^, < ) = sup((𝑀[,)+∞), ℝ^, < ))
49	1	renepnfd 11261	. . . . 5 ⊢ (𝜑 → 𝑀 ≠ +∞)
50		icopnfsup 13826	. . . . 5 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≠ +∞) → sup((𝑀[,)+∞), ℝ^*, < ) = +∞)
51	2, 49, 50	syl2anc 584	. . . 4 ⊢ (𝜑 → sup((𝑀[,)+∞), ℝ^*, < ) = +∞)
52	48, 51	eqtrd 2772	. . 3 ⊢ (𝜑 → sup(𝑍, ℝ^*, < ) = +∞)
53	44, 46, 39, 52	liminfval2 44470	. 2 ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = sup(((𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝑍) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍), ℝ^*, < ))
54		eqid 2732	. . 3 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
55	54, 45, 39, 52	liminfval2 44470	. 2 ⊢ (𝜑 → (lim inf‘𝐹) = sup(((𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) “ 𝑍), ℝ^*, < ))
56	43, 53, 55	3eqtr4d 2782	1 ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ↾ cres 5677 “ cima 5678 ‘cfv 6540 (class class class)co 7405 supcsup 9431 infcinf 9432 ℝcr 11105 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 [,)cico 13322 lim infclsi 44453

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-ico 13326 df-liminf 44454

This theorem is referenced by: liminfresre 44481 liminfresicompt 44482 liminfresuz 44486

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