Theorem liminfequzmpt2 45768

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 45392	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐴)
5	4	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐴)
6		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ (ℤ≥‘𝐾))
7	5, 6	sseldd 3959	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐴)
8		liminfequzmpt2.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉)
9	8	elexd 3483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ V)
10	7, 9	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
11		rabid 3437	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↔ (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
12	10, 11	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
13	12	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) → 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}))
14	1, 13	ralrimi 3240	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
15		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗(ℤ≥‘𝐾)
16		nfrab1 3436	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
17	15, 16	dfss3f 3950	. . . . . . . 8 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↔ ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
18	14, 17	sylibr 234	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
19	16, 15	resmptf 6026	. . . . . . 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 2741	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
22	21	fveq2d 6879	. . . 4 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)) = (lim inf‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))))
23	2, 3	eluzelz2d 45388	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
24		eqid 2735	. . . . 5 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾)
25		liminfequzmpt2.o	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
26	2	fvexi 6889	. . . . . . . 8 ⊢ 𝐴 ∈ V
27	25, 26	rabexf 45106	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ∈ V
28	16, 27	mptexf 45209	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
29	28	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
30		eqid 2735	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
31	16, 30	dmmptssf 45204	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
32	25	ssrab2f 45089	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ 𝐴
33		uzssz 12871	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
34	2, 33	eqsstri 4005	. . . . . . . 8 ⊢ 𝐴 ⊆ ℤ
35	32, 34	sstri 3968	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ ℤ
36	31, 35	sstri 3968	. . . . . 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 45764	. . . 4 ⊢ (𝜑 → (lim inf‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)))
39	22, 38	eqtr2d 2771	. . 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 45392	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐵)
43	42	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐵)
44	43, 6	sseldd 3959	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐵)
45	44, 9	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
46		rabid 3437	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
47	45, 46	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
48	47	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) → 𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}))
49	1, 48	ralrimi 3240	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
50		nfrab1 3436	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
51	15, 50	dfss3f 3950	. . . . . . . 8 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
52	49, 51	sylibr 234	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
53	50, 15	resmptf 6026	. . . . . . 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 2741	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
56	55	fveq2d 6879	. . . 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 6889	. . . . . . . 8 ⊢ 𝐵 ∈ V
59	57, 58	rabexf 45106	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ∈ V
60	50, 59	mptexf 45209	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
62		eqid 2735	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
63	50, 62	dmmptssf 45204	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
64	57	ssrab2f 45089	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ 𝐵
65		uzssz 12871	. . . . . . . . 9 ⊢ (ℤ≥‘𝑁) ⊆ ℤ
66	40, 65	eqsstri 4005	. . . . . . . 8 ⊢ 𝐵 ⊆ ℤ
67	64, 66	sstri 3968	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ ℤ
68	63, 67	sstri 3968	. . . . . 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 45764	. . . 4 ⊢ (𝜑 → (lim inf‘((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
71	56, 70	eqtr2d 2771	. . 3 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim inf‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)))
72	39, 71	eqtr4d 2773	. 2 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim inf‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
73		eqid 2735	. . . . 5 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
74	25, 73	mptssid 45213	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
75	74	fveq2i 6878	. . 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 2735	. . . . 5 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
78	57, 77	mptssid 45213	. . . 4 ⊢ (𝑗 ∈ 𝐵 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
79	78	fveq2i 6878	. . 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 2780	1 ⊢ (𝜑 → (lim inf‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ 𝐵 ↦ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2883  ∀wral 3051  {crab 3415  Vcvv 3459   ⊆ wss 3926   ↦ cmpt 5201  dom cdm 5654   ↾ cres 5656  ‘cfv 6530  ℤcz 12586  ℤ_≥cuz 12850  lim infclsi 45728

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9452  df-inf 9453  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-n0 12500  df-z 12587  df-uz 12851  df-q 12963  df-ico 13366  df-liminf 45729

This theorem is referenced by:  smfliminfmpt  46809