limsupequzlem - Mathbox for Glauco Siliprandi

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Theorem limsupequzlem 45718

Description: Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
limsupequzlem.1	⊢ Ⅎ𝑘𝜑
limsupequzlem.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
limsupequzlem.4	⊢ (𝜑 → 𝐹 Fn (ℤ_≥‘𝑀))
limsupequzlem.5	⊢ (𝜑 → 𝑁 ∈ ℤ)
limsupequzlem.6	⊢ (𝜑 → 𝐺 Fn (ℤ_≥‘𝑁))
limsupequzlem.7	⊢ (𝜑 → 𝐾 ∈ ℤ)
limsupequzlem.8	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘))

**Assertion**
Ref	Expression
limsupequzlem	⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺))

Distinct variable groups: 𝑘,𝐹 𝑘,𝐺 𝑘,𝐾 𝑘,𝑀 𝑘,𝑁 Allowed substitution hint: 𝜑(𝑘)

**Proof of Theorem limsupequzlem**
Step	Hyp	Ref	Expression
1		limsupequzlem.1	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		eqid 2736	. . . . . . 7 ⊢ (ℤ_≥‘𝐾) = (ℤ_≥‘𝐾)
3		limsupequzlem.7	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝐾 ∈ ℤ)
5		eluzelz 12867	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )) → 𝑘 ∈ ℤ)
6	5	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝑘 ∈ ℤ)
7	3	zred 12702	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝐾 ∈ ℝ)
9	8	rexrd 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → 𝐾 ∈ ℝ^)
10		zssxr 45391	. . . . . . . . . 10 ⊢ ℤ ⊆ ℝ^*
11		limsupequzlem.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		limsupequzlem.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		tpssi 4819	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → {𝑀, 𝑁, 𝐾} ⊆ ℤ)
14	11, 12, 3, 13	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ⊆ ℤ)
15		xrltso 13162	. . . . . . . . . . . . 13 ⊢ < Or ℝ^*
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → < Or ℝ^*)
17		tpfi 9342	. . . . . . . . . . . . 13 ⊢ {𝑀, 𝑁, 𝐾} ∈ Fin
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ∈ Fin)
19	11	tpnzd 4761	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ≠ ∅)
20	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℤ ⊆ ℝ^*)
21	14, 20	sstrd 3974	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ⊆ ℝ^*)
22		fisupcl 9487	. . . . . . . . . . . 12 ⊢ (( < Or ℝ^* ∧ ({𝑀, 𝑁, 𝐾} ∈ Fin ∧ {𝑀, 𝑁, 𝐾} ≠ ∅ ∧ {𝑀, 𝑁, 𝐾} ⊆ ℝ^)) → sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) ∈ {𝑀, 𝑁, 𝐾})
23	16, 18, 19, 21, 22	syl13anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ {𝑀, 𝑁, 𝐾})
24	14, 23	sseldd 3964	. . . . . . . . . 10 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ ℤ)
25	10, 24	sselid 3961	. . . . . . . . 9 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) ∈ ℝ^)
26	25	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) ∈ ℝ^*)
27		eluzelre 12868	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )) → 𝑘 ∈ ℝ)
28	27	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝑘 ∈ ℝ)
29	28	rexrd 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → 𝑘 ∈ ℝ^)
30		tpid3g 4753	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ {𝑀, 𝑁, 𝐾})
31	3, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ {𝑀, 𝑁, 𝐾})
32		eqid 2736	. . . . . . . . . 10 ⊢ sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) = sup({𝑀, 𝑁, 𝐾}, ℝ^, < )
33	21, 31, 32	supxrubd 45104	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → 𝐾 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))
35		eluzle 12870	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )) → sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) ≤ 𝑘)
36	35	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) ≤ 𝑘)
37	9, 26, 29, 34, 36	xrletrd 13183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝐾 ≤ 𝑘)
38	2, 4, 6, 37	eluzd 45403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝑘 ∈ (ℤ_≥‘𝐾))
39		limsupequzlem.8	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘))
40	38, 39	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → (𝐹‘𝑘) = (𝐺‘𝑘))
41	1, 40	ralrimia 3245	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))(𝐹‘𝑘) = (𝐺‘𝑘))
42		limsupequzlem.4	. . . . 5 ⊢ (𝜑 → 𝐹 Fn (ℤ_≥‘𝑀))
43		limsupequzlem.6	. . . . 5 ⊢ (𝜑 → 𝐺 Fn (ℤ_≥‘𝑁))
44		eqid 2736	. . . . . . 7 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
45		tpid1g 4750	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ {𝑀, 𝑁, 𝐾})
46	11, 45	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ {𝑀, 𝑁, 𝐾})
47	21, 46, 32	supxrubd 45104	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))
48	44, 11, 24, 47	eluzd 45403	. . . . . 6 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ (ℤ_≥‘𝑀))
49		uzss 12880	. . . . . 6 ⊢ (sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )) ⊆ (ℤ_≥‘𝑀))
50	48, 49	syl 17	. . . . 5 ⊢ (𝜑 → (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )) ⊆ (ℤ_≥‘𝑀))
51		eqid 2736	. . . . . . 7 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
52		tpid2g 4752	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ {𝑀, 𝑁, 𝐾})
53	12, 52	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ {𝑀, 𝑁, 𝐾})
54	21, 53, 32	supxrubd 45104	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))
55	51, 12, 24, 54	eluzd 45403	. . . . . 6 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ (ℤ_≥‘𝑁))
56		uzss 12880	. . . . . 6 ⊢ (sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )) ⊆ (ℤ_≥‘𝑁))
57	55, 56	syl 17	. . . . 5 ⊢ (𝜑 → (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )) ⊆ (ℤ_≥‘𝑁))
58		fvreseq0 7033	. . . . 5 ⊢ (((𝐹 Fn (ℤ_≥‘𝑀) ∧ 𝐺 Fn (ℤ_≥‘𝑁)) ∧ ((ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )) ⊆ (ℤ_≥‘𝑀) ∧ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )) ⊆ (ℤ_≥‘𝑁))) → ((𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) = (𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) ↔ ∀𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))(𝐹‘𝑘) = (𝐺‘𝑘)))
59	42, 43, 50, 57, 58	syl22anc 838	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) = (𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) ↔ ∀𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))(𝐹‘𝑘) = (𝐺‘𝑘)))
60	41, 59	mpbird 257	. . 3 ⊢ (𝜑 → (𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) = (𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))))
61	60	fveq2d 6885	. 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )))) = (lim sup‘(𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )))))
62		eqid 2736	. . 3 ⊢ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )) = (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))
63		fvexd 6896	. . . 4 ⊢ (𝜑 → (ℤ_≥‘𝑀) ∈ V)
64	42, 63	fnexd 7215	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
65	42	fndmd 6648	. . . 4 ⊢ (𝜑 → dom 𝐹 = (ℤ_≥‘𝑀))
66		uzssz 12878	. . . 4 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
67	65, 66	eqsstrdi 4008	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)
68	24, 62, 64, 67	limsupresuz2 45705	. 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )))) = (lim sup‘𝐹))
69		fvexd 6896	. . . 4 ⊢ (𝜑 → (ℤ_≥‘𝑁) ∈ V)
70	43, 69	fnexd 7215	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
71	43	fndmd 6648	. . . 4 ⊢ (𝜑 → dom 𝐺 = (ℤ_≥‘𝑁))
72		uzssz 12878	. . . 4 ⊢ (ℤ_≥‘𝑁) ⊆ ℤ
73	71, 72	eqsstrdi 4008	. . 3 ⊢ (𝜑 → dom 𝐺 ⊆ ℤ)
74	24, 62, 70, 73	limsupresuz2 45705	. 2 ⊢ (𝜑 → (lim sup‘(𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )))) = (lim sup‘𝐺))
75	61, 68, 74	3eqtr3d 2779	1 ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 Vcvv 3464 ⊆ wss 3931 ∅c0 4313 {ctp 4610 class class class wbr 5124 Or wor 5565 dom cdm 5659 ↾ cres 5661 Fn wfn 6531 ‘cfv 6536 Fincfn 8964 supcsup 9457 ℝcr 11133 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 ℤcz 12593 ℤ_≥cuz 12857 lim supclsp 15491

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-n0 12507 df-z 12594 df-uz 12858 df-q 12970 df-ico 13373 df-limsup 15492

This theorem is referenced by: limsupequz 45719

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