Theorem liminfequzmpt2 40023

Description: Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
liminfequzmpt2.j	⊢ Ⅎ𝑗𝜑
liminfequzmpt2.o	⊢ Ⅎ𝑗𝐴
liminfequzmpt2.p	⊢ Ⅎ𝑗𝐵
liminfequzmpt2.a	⊢ 𝐴 = (ℤ≥‘𝑀)
liminfequzmpt2.b	⊢ 𝐵 = (ℤ≥‘𝑁)
liminfequzmpt2.k	⊢ (𝜑 → 𝐾 ∈ 𝐴)
liminfequzmpt2.e	⊢ (𝜑 → 𝐾 ∈ 𝐵)
liminfequzmpt2.c	⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
liminfequzmpt2	⊢ (𝜑 → (lim inf‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ 𝐵 ↦ 𝐶)))

Distinct variable group:   𝑗,𝐾

Allowed substitution hints:   𝜑(𝑗)   𝐴(𝑗)   𝐵(𝑗)   𝐶(𝑗)   𝑀(𝑗)   𝑁(𝑗)   𝑉(𝑗)

Proof of Theorem liminfequzmpt2

Step	Hyp	Ref	Expression
1		liminfequzmpt2.j	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
2		liminfequzmpt2.a	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = (ℤ≥‘𝑀)
3		liminfequzmpt2.k	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝐴)
4	2, 3	uzssd2 39644	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐴)
5	4	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐴)
6		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ (ℤ≥‘𝐾))
7	5, 6	sseldd 3604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐴)
8		liminfequzmpt2.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉)
9	8	elexd 3214	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ V)
10	7, 9	jca 554	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
11		rabid 3116	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↔ (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
12	10, 11	sylibr 224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
13	12	ex 450	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) → 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}))
14	1, 13	ralrimi 2957	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
15		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑗(ℤ≥‘𝐾)
16		nfrab1 3122	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
17	15, 16	dfss3f 3595	. . . . . . . 8 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↔ ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
18	14, 17	sylibr 224	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
19	16, 15	resmptf 5451	. . . . . . 7 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} → ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)) = (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶))
20	18, 19	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)) = (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶))
21	20	eqcomd 2628	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
22	21	fveq2d 6195	. . . 4 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)) = (lim inf‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))))
23	2, 3	eluzelz2d 39640	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
24		eqid 2622	. . . . 5 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾)
25		liminfequzmpt2.o	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
26	2	fvexi 6202	. . . . . . . 8 ⊢ 𝐴 ∈ V
27	25, 26	rabexf 39319	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ∈ V
28	16, 27	mptexf 39444	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
29	28	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
30		eqid 2622	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
31	16, 30	dmmptssf 39438	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
32	25	ssrab2f 39300	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ 𝐴
33		uzssz 11707	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
34	2, 33	eqsstri 3635	. . . . . . . 8 ⊢ 𝐴 ⊆ ℤ
35	32, 34	sstri 3612	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ ℤ
36	31, 35	sstri 3612	. . . . . 6 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ ℤ
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ ℤ)
38	23, 24, 29, 37	liminfresuz2 40019	. . . 4 ⊢ (𝜑 → (lim inf‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)))
39	22, 38	eqtr2d 2657	. . 3 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim inf‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)))
40		liminfequzmpt2.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (ℤ≥‘𝑁)
41		liminfequzmpt2.e	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝐵)
42	40, 41	uzssd2 39644	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐵)
43	42	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐵)
44	43, 6	sseldd 3604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐵)
45	44, 9	jca 554	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
46		rabid 3116	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
47	45, 46	sylibr 224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
48	47	ex 450	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) → 𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}))
49	1, 48	ralrimi 2957	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
50		nfrab1 3122	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
51	15, 50	dfss3f 3595	. . . . . . . 8 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
52	49, 51	sylibr 224	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
53	50, 15	resmptf 5451	. . . . . . 7 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} → ((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)) = (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶))
54	52, 53	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)) = (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶))
55	54	eqcomd 2628	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
56	55	fveq2d 6195	. . . 4 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)) = (lim inf‘((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))))
57		liminfequzmpt2.p	. . . . . . . 8 ⊢ Ⅎ𝑗𝐵
58	40	fvexi 6202	. . . . . . . 8 ⊢ 𝐵 ∈ V
59	57, 58	rabexf 39319	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ∈ V
60	50, 59	mptexf 39444	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
62		eqid 2622	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
63	50, 62	dmmptssf 39438	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
64	57	ssrab2f 39300	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ 𝐵
65		uzssz 11707	. . . . . . . . 9 ⊢ (ℤ≥‘𝑁) ⊆ ℤ
66	40, 65	eqsstri 3635	. . . . . . . 8 ⊢ 𝐵 ⊆ ℤ
67	64, 66	sstri 3612	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ ℤ
68	63, 67	sstri 3612	. . . . . 6 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ ℤ
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → dom (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ ℤ)
70	23, 24, 61, 69	liminfresuz2 40019	. . . 4 ⊢ (𝜑 → (lim inf‘((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
71	56, 70	eqtr2d 2657	. . 3 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim inf‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)))
72	39, 71	eqtr4d 2659	. 2 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
73		eqid 2622	. . . . 5 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
74	25, 73	mptssid 39450	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
75	74	fveq2i 6194	. . 3 ⊢ (lim inf‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶))
76	75	a1i 11	. 2 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)))
77		eqid 2622	. . . . 5 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
78	57, 77	mptssid 39450	. . . 4 ⊢ (𝑗 ∈ 𝐵 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
79	78	fveq2i 6194	. . 3 ⊢ (lim inf‘(𝑗 ∈ 𝐵 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶))
80	79	a1i 11	. 2 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ 𝐵 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
81	72, 76, 80	3eqtr4d 2666	1 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ 𝐵 ↦ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751  ∀wral 2912  {crab 2916  Vcvv 3200   ⊆ wss 3574   ↦ cmpt 4729  dom cdm 5114   ↾ cres 5116  ‘cfv 5888  ℤcz 11377  ℤ_≥cuz 11687  lim infclsi 39983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ico 12181  df-liminf 39984

This theorem is referenced by:  smfliminfmpt  41038