Theorem limsupequzmpt2 44424

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 44117	. . . . . . . . . . . . . 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 3983	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐴)
8		limsupequzmpt2.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉)
9	8	elexd 3494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ V)
10	7, 9	jca 512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐴 ∧ 𝐶 ∈ V))
11		rabid 3452	. . . . . . . . . . 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 3254	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V})
15		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑗(ℤ≥‘𝐾)
16		nfrab1 3451	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
17	15, 16	dfss3f 3973	. . . . . . . 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 6039	. . . . . . 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 2738	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
22	21	fveq2d 6895	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)) = (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))))
23	2, 3	eluzelz2d 44113	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
24		eqid 2732	. . . . 5 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾)
25		limsupequzmpt2.o	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
26	2	fvexi 6905	. . . . . . . 8 ⊢ 𝐴 ∈ V
27	25, 26	rabexf 43813	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ∈ V
28	16, 27	mptexf 43930	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
29	28	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
30		eqid 2732	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
31	16, 30	dmmptssf 43924	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
32	25	ssrab2f 43796	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ 𝐴
33		uzssz 12842	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
34	2, 33	eqsstri 4016	. . . . . . . 8 ⊢ 𝐴 ⊆ ℤ
35	32, 34	sstri 3991	. . . . . . 7 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ⊆ ℤ
36	31, 35	sstri 3991	. . . . . 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 44415	. . . 4 ⊢ (𝜑 → (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)))
39	22, 38	eqtr2d 2773	. . 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 44117	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘𝐾) ⊆ 𝐵)
43	42	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (ℤ≥‘𝐾) ⊆ 𝐵)
44	43, 6	sseldd 3983	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝑗 ∈ 𝐵)
45	44, 9	jca 512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝑗 ∈ 𝐵 ∧ 𝐶 ∈ V))
46		rabid 3452	. . . . . . . . . . 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 3254	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V})
50		nfrab1 3451	. . . . . . . . 9 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
51	15, 50	dfss3f 3973	. . . . . . . 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 6039	. . . . . . 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 2738	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶) = ((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾)))
56	55	fveq2d 6895	. . . 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 6905	. . . . . . . 8 ⊢ 𝐵 ∈ V
59	57, 58	rabexf 43813	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ∈ V
60	50, 59	mptexf 43930	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ∈ V)
62		eqid 2732	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
63	50, 62	dmmptssf 43924	. . . . . . 7 ⊢ dom (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ⊆ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
64	57	ssrab2f 43796	. . . . . . . 8 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ 𝐵
65		uzssz 12842	. . . . . . . . 9 ⊢ (ℤ≥‘𝑁) ⊆ ℤ
66	40, 65	eqsstri 4016	. . . . . . . 8 ⊢ 𝐵 ⊆ ℤ
67	64, 66	sstri 3991	. . . . . . 7 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ⊆ ℤ
68	63, 67	sstri 3991	. . . . . 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 44415	. . . 4 ⊢ (𝜑 → (lim sup‘((𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶) ↾ (ℤ≥‘𝐾))) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
71	56, 70	eqtr2d 2773	. . 3 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim sup‘(𝑗 ∈ (ℤ≥‘𝐾) ↦ 𝐶)))
72	39, 71	eqtr4d 2775	. 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)) = (lim sup‘(𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)))
73		eqid 2732	. . . . 5 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V}
74	25, 73	mptssid 43934	. . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐶 ∈ V} ↦ 𝐶)
75	74	fveq2i 6894	. . 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 2732	. . . . 5 ⊢ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} = {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V}
78	57, 77	mptssid 43934	. . . 4 ⊢ (𝑗 ∈ 𝐵 ↦ 𝐶) = (𝑗 ∈ {𝑗 ∈ 𝐵 ∣ 𝐶 ∈ V} ↦ 𝐶)
79	78	fveq2i 6894	. . 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 2782	1 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  Ⅎwnfc 2883  ∀wral 3061  {crab 3432  Vcvv 3474   ⊆ wss 3948   ↦ cmpt 5231  dom cdm 5676   ↾ cres 5678  ‘cfv 6543  ℤcz 12557  ℤ_≥cuz 12821  lim supclsp 15413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-q 12932  df-ico 13329  df-limsup 15414

This theorem is referenced by:  smflimsupmpt  45535