Theorem limsupequzmpt2 40851

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
limsupequzmpt2.j	⊢ Ⅎ𝑗𝜑
limsupequzmpt2.o	⊢ Ⅎ𝑗𝐴
limsupequzmpt2.p	⊢ Ⅎ𝑗𝐵
limsupequzmpt2.a	⊢ 𝐴 = (ℤ≥‘𝑀)
limsupequzmpt2.b	⊢ 𝐵 = (ℤ≥‘𝑁)
limsupequzmpt2.k	⊢ (𝜑 → 𝐾 ∈ 𝐴)
limsupequzmpt2.e	⊢ (𝜑 → 𝐾 ∈ 𝐵)
limsupequzmpt2.c	⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
limsupequzmpt2	⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(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 limsupequzmpt2

Step	Hyp	Ref	Expression
1		limsupequzmpt2.j	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
2		limsupequzmpt2.a	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = (ℤ≥‘𝑀)
3		limsupequzmpt2.k	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝐴)
4	2, 3	uzssd2 40543	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐴)
5	4	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐴)
6		simpr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ (ℤ≥‘𝐾))
7	5, 6	sseldd 3821	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐴)
8		limsupequzmpt2.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉)
9	8	elexd 3415	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ V)
10	7, 9	jca 507	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
11		rabid 3301	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↔ (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
12	10, 11	sylibr 226	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
13	12	ex 403	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) → 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}))
14	1, 13	ralrimi 3138	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
15		nfcv 2933	. . . . . . . . 9 ⊢ Ⅎ𝑗(ℤ≥‘𝐾)
16		nfrab1 3308	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
17	15, 16	dfss3f 3812	. . . . . . . 8 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↔ ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
18	14, 17	sylibr 226	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
19	16, 15	resmptf 5701	. . . . . . 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 2783	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
22	21	fveq2d 6450	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)) = (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))))
23	2, 3	eluzelz2d 40539	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
24		eqid 2777	. . . . 5 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾)
25		limsupequzmpt2.o	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
26	2	fvexi 6460	. . . . . . . 8 ⊢ 𝐴 ∈ V
27	25, 26	rabexf 40239	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ∈ V
28	16, 27	mptexf 40353	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
29	28	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
30		eqid 2777	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
31	16, 30	dmmptssf 40348	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
32	25	ssrab2f 40222	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ 𝐴
33		uzssz 12012	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
34	2, 33	eqsstri 3853	. . . . . . . 8 ⊢ 𝐴 ⊆ ℤ
35	32, 34	sstri 3829	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ ℤ
36	31, 35	sstri 3829	. . . . . 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	limsupresuz2 40842	. . . 4 ⊢ (𝜑 → (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)))
39	22, 38	eqtr2d 2814	. . 3 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim sup‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)))
40		limsupequzmpt2.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (ℤ≥‘𝑁)
41		limsupequzmpt2.e	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝐵)
42	40, 41	uzssd2 40543	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐵)
43	42	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐵)
44	43, 6	sseldd 3821	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐵)
45	44, 9	jca 507	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
46		rabid 3301	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
47	45, 46	sylibr 226	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
48	47	ex 403	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) → 𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}))
49	1, 48	ralrimi 3138	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
50		nfrab1 3308	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
51	15, 50	dfss3f 3812	. . . . . . . 8 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
52	49, 51	sylibr 226	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
53	50, 15	resmptf 5701	. . . . . . 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 2783	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
56	55	fveq2d 6450	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)) = (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))))
57		limsupequzmpt2.p	. . . . . . . 8 ⊢ Ⅎ𝑗𝐵
58	40	fvexi 6460	. . . . . . . 8 ⊢ 𝐵 ∈ V
59	57, 58	rabexf 40239	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ∈ V
60	50, 59	mptexf 40353	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
62		eqid 2777	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
63	50, 62	dmmptssf 40348	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
64	57	ssrab2f 40222	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ 𝐵
65		uzssz 12012	. . . . . . . . 9 ⊢ (ℤ≥‘𝑁) ⊆ ℤ
66	40, 65	eqsstri 3853	. . . . . . . 8 ⊢ 𝐵 ⊆ ℤ
67	64, 66	sstri 3829	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ ℤ
68	63, 67	sstri 3829	. . . . . 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	limsupresuz2 40842	. . . 4 ⊢ (𝜑 → (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
71	56, 70	eqtr2d 2814	. . 3 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim sup‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)))
72	39, 71	eqtr4d 2816	. 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
73		eqid 2777	. . . . 5 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
74	25, 73	mptssid 40357	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
75	74	fveq2i 6449	. . 3 ⊢ (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶))
76	75	a1i 11	. 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)))
77		eqid 2777	. . . . 5 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
78	57, 77	mptssid 40357	. . . 4 ⊢ (𝑗 ∈ 𝐵 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
79	78	fveq2i 6449	. . 3 ⊢ (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶))
80	79	a1i 11	. 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
81	72, 76, 80	3eqtr4d 2823	1 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601  Ⅎwnf 1827   ∈ wcel 2106  Ⅎwnfc 2918  ∀wral 3089  {crab 3093  Vcvv 3397   ⊆ wss 3791   ↦ cmpt 4965  dom cdm 5355   ↾ cres 5357  ‘cfv 6135  ℤcz 11728  ℤ_≥cuz 11992  lim supclsp 14609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-sup 8636  df-inf 8637  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-q 12096  df-ico 12493  df-limsup 14610

This theorem is referenced by:  smflimsupmpt  41955