limsupequzlem - Mathbox for Glauco Siliprandi

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Theorem limsupequzlem 45908

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 2734	. . . . . . 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 12759	. . . . . . . 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 12594	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝐾 ∈ ℝ)
9	8	rexrd 11180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → 𝐾 ∈ ℝ^)
10		zssxr 45583	. . . . . . . . . 10 ⊢ ℤ ⊆ ℝ^*
11		limsupequzlem.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		limsupequzlem.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		tpssi 4792	. . . . . . . . . . . 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 13053	. . . . . . . . . . . . 13 ⊢ < Or ℝ^*
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → < Or ℝ^*)
17		tpfi 9224	. . . . . . . . . . . . 13 ⊢ {𝑀, 𝑁, 𝐾} ∈ Fin
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ∈ Fin)
19	11	tpnzd 4735	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ≠ ∅)
20	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℤ ⊆ ℝ^*)
21	14, 20	sstrd 3942	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ⊆ ℝ^*)
22		fisupcl 9371	. . . . . . . . . . . 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 3932	. . . . . . . . . 10 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ ℤ)
25	10, 24	sselid 3929	. . . . . . . . 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 12760	. . . . . . . . . 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 11180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → 𝑘 ∈ ℝ^)
30		tpid3g 4727	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ {𝑀, 𝑁, 𝐾})
31	3, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ {𝑀, 𝑁, 𝐾})
32		eqid 2734	. . . . . . . . . 10 ⊢ sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) = sup({𝑀, 𝑁, 𝐾}, ℝ^, < )
33	21, 31, 32	supxrubd 45299	. . . . . . . . 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 12762	. . . . . . . . 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 13074	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝐾 ≤ 𝑘)
38	2, 4, 6, 37	eluzd 45595	. . . . . 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 3233	. . . 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 2734	. . . . . . 7 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
45		tpid1g 4724	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ {𝑀, 𝑁, 𝐾})
46	11, 45	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ {𝑀, 𝑁, 𝐾})
47	21, 46, 32	supxrubd 45299	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))
48	44, 11, 24, 47	eluzd 45595	. . . . . 6 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ (ℤ_≥‘𝑀))
49		uzss 12772	. . . . . 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 2734	. . . . . . 7 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
52		tpid2g 4726	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ {𝑀, 𝑁, 𝐾})
53	12, 52	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ {𝑀, 𝑁, 𝐾})
54	21, 53, 32	supxrubd 45299	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))
55	51, 12, 24, 54	eluzd 45595	. . . . . 6 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ (ℤ_≥‘𝑁))
56		uzss 12772	. . . . . 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 6981	. . . . 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 6836	. 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )))) = (lim sup‘(𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )))))
62		eqid 2734	. . 3 ⊢ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )) = (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))
63		fvexd 6847	. . . 4 ⊢ (𝜑 → (ℤ_≥‘𝑀) ∈ V)
64	42, 63	fnexd 7162	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
65	42	fndmd 6595	. . . 4 ⊢ (𝜑 → dom 𝐹 = (ℤ_≥‘𝑀))
66		uzssz 12770	. . . 4 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
67	65, 66	eqsstrdi 3976	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)
68	24, 62, 64, 67	limsupresuz2 45895	. 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )))) = (lim sup‘𝐹))
69		fvexd 6847	. . . 4 ⊢ (𝜑 → (ℤ_≥‘𝑁) ∈ V)
70	43, 69	fnexd 7162	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
71	43	fndmd 6595	. . . 4 ⊢ (𝜑 → dom 𝐺 = (ℤ_≥‘𝑁))
72		uzssz 12770	. . . 4 ⊢ (ℤ_≥‘𝑁) ⊆ ℤ
73	71, 72	eqsstrdi 3976	. . 3 ⊢ (𝜑 → dom 𝐺 ⊆ ℤ)
74	24, 62, 70, 73	limsupresuz2 45895	. 2 ⊢ (𝜑 → (lim sup‘(𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )))) = (lim sup‘𝐺))
75	61, 68, 74	3eqtr3d 2777	1 ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 Ⅎwnf 1784 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 Vcvv 3438 ⊆ wss 3899 ∅c0 4283 {ctp 4582 class class class wbr 5096 Or wor 5529 dom cdm 5622 ↾ cres 5624 Fn wfn 6485 ‘cfv 6490 Fincfn 8881 supcsup 9341 ℝcr 11023 ℝ^*cxr 11163 < clt 11164 ≤ cle 11165 ℤcz 12486 ℤ_≥cuz 12749 lim supclsp 15391

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-n0 12400 df-z 12487 df-uz 12750 df-q 12860 df-ico 13265 df-limsup 15392

This theorem is referenced by: limsupequz 45909

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