limsupequzlem - Mathbox for Glauco Siliprandi

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Theorem limsupequzlem 45643

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 2740	. . . . . . 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 12913	. . . . . . . 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 12747	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝐾 ∈ ℝ)
9	8	rexrd 11340	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → 𝐾 ∈ ℝ^)
10		zssxr 45312	. . . . . . . . . 10 ⊢ ℤ ⊆ ℝ^*
11		limsupequzlem.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		limsupequzlem.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		tpssi 4863	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → {𝑀, 𝑁, 𝐾} ⊆ ℤ)
14	11, 12, 3, 13	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ⊆ ℤ)
15		xrltso 13203	. . . . . . . . . . . . 13 ⊢ < Or ℝ^*
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → < Or ℝ^*)
17		tpfi 9393	. . . . . . . . . . . . 13 ⊢ {𝑀, 𝑁, 𝐾} ∈ Fin
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ∈ Fin)
19	11	tpnzd 4805	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ≠ ∅)
20	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℤ ⊆ ℝ^*)
21	14, 20	sstrd 4019	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑀, 𝑁, 𝐾} ⊆ ℝ^*)
22		fisupcl 9538	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ {𝑀, 𝑁, 𝐾})
24	14, 23	sseldd 4009	. . . . . . . . . 10 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ ℤ)
25	10, 24	sselid 4006	. . . . . . . . 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 12914	. . . . . . . . . 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 11340	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))) → 𝑘 ∈ ℝ^)
30		tpid3g 4797	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ {𝑀, 𝑁, 𝐾})
31	3, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ {𝑀, 𝑁, 𝐾})
32		eqid 2740	. . . . . . . . . 10 ⊢ sup({𝑀, 𝑁, 𝐾}, ℝ^, < ) = sup({𝑀, 𝑁, 𝐾}, ℝ^, < )
33	21, 31, 32	supxrubd 45015	. . . . . . . . 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 12916	. . . . . . . . 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 13224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → 𝐾 ≤ 𝑘)
38	2, 4, 6, 37	eluzd 45324	. . . . . 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 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))) → (𝐹‘𝑘) = (𝐺‘𝑘))
41	1, 40	ralrimia 3264	. . . 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 2740	. . . . . . 7 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
45		tpid1g 4794	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ {𝑀, 𝑁, 𝐾})
46	11, 45	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ {𝑀, 𝑁, 𝐾})
47	21, 46, 32	supxrubd 45015	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))
48	44, 11, 24, 47	eluzd 45324	. . . . . 6 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ (ℤ_≥‘𝑀))
49		uzss 12926	. . . . . 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 2740	. . . . . . 7 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
52		tpid2g 4796	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ {𝑀, 𝑁, 𝐾})
53	12, 52	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ {𝑀, 𝑁, 𝐾})
54	21, 53, 32	supxrubd 45015	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≤ sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ))
55	51, 12, 24, 54	eluzd 45324	. . . . . 6 ⊢ (𝜑 → sup({𝑀, 𝑁, 𝐾}, ℝ^*, < ) ∈ (ℤ_≥‘𝑁))
56		uzss 12926	. . . . . 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 7071	. . . . 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 6924	. 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )))) = (lim sup‘(𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )))))
62		eqid 2740	. . 3 ⊢ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < )) = (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^, < ))
63		fvexd 6935	. . . 4 ⊢ (𝜑 → (ℤ_≥‘𝑀) ∈ V)
64	42, 63	fnexd 7255	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
65	42	fndmd 6684	. . . 4 ⊢ (𝜑 → dom 𝐹 = (ℤ_≥‘𝑀))
66		uzssz 12924	. . . 4 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
67	65, 66	eqsstrdi 4063	. . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)
68	24, 62, 64, 67	limsupresuz2 45630	. 2 ⊢ (𝜑 → (lim sup‘(𝐹 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )))) = (lim sup‘𝐹))
69		fvexd 6935	. . . 4 ⊢ (𝜑 → (ℤ_≥‘𝑁) ∈ V)
70	43, 69	fnexd 7255	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
71	43	fndmd 6684	. . . 4 ⊢ (𝜑 → dom 𝐺 = (ℤ_≥‘𝑁))
72		uzssz 12924	. . . 4 ⊢ (ℤ_≥‘𝑁) ⊆ ℤ
73	71, 72	eqsstrdi 4063	. . 3 ⊢ (𝜑 → dom 𝐺 ⊆ ℤ)
74	24, 62, 70, 73	limsupresuz2 45630	. 2 ⊢ (𝜑 → (lim sup‘(𝐺 ↾ (ℤ_≥‘sup({𝑀, 𝑁, 𝐾}, ℝ^*, < )))) = (lim sup‘𝐺))
75	61, 68, 74	3eqtr3d 2788	1 ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 Vcvv 3488 ⊆ wss 3976 ∅c0 4352 {ctp 4652 class class class wbr 5166 Or wor 5606 dom cdm 5700 ↾ cres 5702 Fn wfn 6568 ‘cfv 6573 Fincfn 9003 supcsup 9509 ℝcr 11183 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 ℤcz 12639 ℤ_≥cuz 12903 lim supclsp 15516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-ico 13413 df-limsup 15517

This theorem is referenced by: limsupequz 45644

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