Theorem limsupequzmpt2 43259

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 42957	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐴)
5	4	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐴)
6		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ (ℤ≥‘𝐾))
7	5, 6	sseldd 3922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐴)
8		limsupequzmpt2.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉)
9	8	elexd 3452	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ V)
10	7, 9	jca 512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
11		rabid 3310	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↔ (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
12	10, 11	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
13	12	ex 413	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) → 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}))
14	1, 13	ralrimi 3141	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
15		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑗(ℤ≥‘𝐾)
16		nfrab1 3317	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
17	15, 16	dfss3f 3912	. . . . . . . 8 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↔ ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
18	14, 17	sylibr 233	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
19	16, 15	resmptf 5947	. . . . . . 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 2744	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
22	21	fveq2d 6778	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)) = (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))))
23	2, 3	eluzelz2d 42953	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
24		eqid 2738	. . . . 5 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾)
25		limsupequzmpt2.o	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
26	2	fvexi 6788	. . . . . . . 8 ⊢ 𝐴 ∈ V
27	25, 26	rabexf 42683	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ∈ V
28	16, 27	mptexf 42781	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
29	28	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
30		eqid 2738	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
31	16, 30	dmmptssf 42775	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
32	25	ssrab2f 42666	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ 𝐴
33		uzssz 12603	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
34	2, 33	eqsstri 3955	. . . . . . . 8 ⊢ 𝐴 ⊆ ℤ
35	32, 34	sstri 3930	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ ℤ
36	31, 35	sstri 3930	. . . . . 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 43250	. . . 4 ⊢ (𝜑 → (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)))
39	22, 38	eqtr2d 2779	. . 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 42957	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐵)
43	42	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐵)
44	43, 6	sseldd 3922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐵)
45	44, 9	jca 512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
46		rabid 3310	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
47	45, 46	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
48	47	ex 413	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) → 𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}))
49	1, 48	ralrimi 3141	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
50		nfrab1 3317	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
51	15, 50	dfss3f 3912	. . . . . . . 8 ⊢ ((ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
52	49, 51	sylibr 233	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
53	50, 15	resmptf 5947	. . . . . . 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 2744	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
56	55	fveq2d 6778	. . . 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 6788	. . . . . . . 8 ⊢ 𝐵 ∈ V
59	57, 58	rabexf 42683	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ∈ V
60	50, 59	mptexf 42781	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
62		eqid 2738	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
63	50, 62	dmmptssf 42775	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
64	57	ssrab2f 42666	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ 𝐵
65		uzssz 12603	. . . . . . . . 9 ⊢ (ℤ≥‘𝑁) ⊆ ℤ
66	40, 65	eqsstri 3955	. . . . . . . 8 ⊢ 𝐵 ⊆ ℤ
67	64, 66	sstri 3930	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ ℤ
68	63, 67	sstri 3930	. . . . . 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 43250	. . . 4 ⊢ (𝜑 → (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
71	56, 70	eqtr2d 2779	. . 3 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim sup‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)))
72	39, 71	eqtr4d 2781	. 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
73		eqid 2738	. . . . 5 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
74	25, 73	mptssid 42785	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
75	74	fveq2i 6777	. . 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 2738	. . . . 5 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
78	57, 77	mptssid 42785	. . . 4 ⊢ (𝑗 ∈ 𝐵 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
79	78	fveq2i 6777	. . 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 2788	1 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064  {crab 3068  Vcvv 3432   ⊆ wss 3887   ↦ cmpt 5157  dom cdm 5589   ↾ cres 5591  ‘cfv 6433  ℤcz 12319  ℤ_≥cuz 12582  lim supclsp 15179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-ico 13085  df-limsup 15180

This theorem is referenced by:  smflimsupmpt  44362