supcnvlimsup - Mathbox for Glauco Siliprandi

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Theorem supcnvlimsup 46633

Description: If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
supcnvlimsup.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
supcnvlimsup.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
supcnvlimsup.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
supcnvlimsup.r	⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

**Assertion**
Ref	Expression
supcnvlimsup	⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^*, < )) ⇝ (lim sup‘𝐹))

Distinct variable groups: 𝑘,𝐹 𝑘,𝑍 Allowed substitution hints: 𝜑(𝑘) 𝑀(𝑘)

Proof of Theorem supcnvlimsup Dummy variables 𝑖 𝑗 𝑥 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supcnvlimsup.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		supcnvlimsup.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		supcnvlimsup.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
4	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
5		id 23	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
6	1, 5	uzssd2 46310	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
7	6	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
8	4, 7	feqresmpt 6950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
9	8	rneqd 5924	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
10	9	supeq1d 9423	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^, < ))
11		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
12		supcnvlimsup.r	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
13	12	renepnfd 11309	. . . . . . . . 9 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
14	11, 1, 3, 13	limsupubuz 46606	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
15	14	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
16		ssralv 4000	. . . . . . . . . 10 ⊢ ((ℤ_≥‘𝑛) ⊆ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
17	6, 16	syl 18	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
18	17	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
19	18	reximdv 3177	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
20	15, 19	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)
21		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
22	1	eluzelz2 46296	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
23		uzid 12927	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
24		ne0i 4287	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘𝑛) ≠ ∅)
25	22, 23, 24	3syl 19	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
26	25	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ≠ ∅)
27	4	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐹:𝑍⟶ℝ)
28	7	sselda 3931	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
29	27, 28	ffvelcdmd 7081	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
30	21, 26, 29	supxrre3rnmpt 46322	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
31	20, 30	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^*, < ) ∈ ℝ)
32	10, 31	eqeltrd 2860	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ) ∈ ℝ)
33	32	fmpttd 7111	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )):𝑍⟶ℝ)
34		eqid 2760	. . . . . . . 8 ⊢ (ℤ_≥‘𝑖) = (ℤ_≥‘𝑖)
35	1	eluzelz2 46296	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℤ)
36	35	peano2zd 12753	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ ℤ)
37	35	zred 12750	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℝ)
38		lep1 12105	. . . . . . . . 9 ⊢ (𝑖 ∈ ℝ → 𝑖 ≤ (𝑖 + 1))
39	37, 38	syl 18	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ≤ (𝑖 + 1))
40	34, 35, 36, 39	eluzd 46302	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ (ℤ_≥‘𝑖))
41		uzss 12935	. . . . . . 7 ⊢ ((𝑖 + 1) ∈ (ℤ_≥‘𝑖) → (ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖))
42		ssres2 5999	. . . . . . 7 ⊢ ((ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖) → (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ_≥‘𝑖)))
43		rnss 5925	. . . . . . 7 ⊢ ((𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ_≥‘𝑖)) → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
44	40, 41, 42, 43	4syl 20	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
45	44	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
46		rnresss 6012	. . . . . . . 8 ⊢ ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ran 𝐹
47	46	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ran 𝐹)
48	3	frnd 6714	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
49	48	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
50	47, 49	sstrd 3941	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ)
51		ressxr 11302	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
52	51	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ℝ ⊆ ℝ^*)
53	50, 52	sstrd 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
54		supxrss 13409	. . . . 5 ⊢ ((ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)) ∧ ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^) → sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
55	45, 53, 54	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
56		eqidd 2761	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )))
57		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑖 + 1) → (ℤ_≥‘𝑛) = (ℤ_≥‘(𝑖 + 1)))
58	57	reseq2d 5974	. . . . . . . . . 10 ⊢ (𝑛 = (𝑖 + 1) → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))))
59	58	rneqd 5924	. . . . . . . . 9 ⊢ (𝑛 = (𝑖 + 1) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))))
60	59	supeq1d 9423	. . . . . . . 8 ⊢ (𝑛 = (𝑖 + 1) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
61	60	adantl 487	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = (𝑖 + 1)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
62	1	peano2uzs 12976	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ 𝑍)
63		xrltso 13217	. . . . . . . . 9 ⊢ < Or ℝ^*
64	63	supex 9441	. . . . . . . 8 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^*, < ) ∈ V
65	64	a1i 11	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^*, < ) ∈ V)
66	56, 61, 62, 65	fvmptd 6997	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
67	66	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
68		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑖))
69	68	reseq2d 5974	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑖)))
70	69	rneqd 5924	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑖)))
71	70	supeq1d 9423	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
72	71	adantl 487	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = 𝑖) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
73		id 23	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ 𝑍)
74	63	supex 9441	. . . . . . . 8 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ) ∈ V
75	74	a1i 11	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ) ∈ V)
76	56, 72, 73, 75	fvmptd 6997	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
77	76	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
78	67, 77	breq12d 5116	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) ↔ sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )))
79	55, 78	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖))
80		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑗𝐹
81	3	frexr 46279	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
82	80, 2, 1, 81	limsupre3uz 46629	. . . . . . 7 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥)))
83	12, 82	mpbid 235	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥))
84	83	simpld 500	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗))
85		simp-4r 796	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ)
86	85	rexrd 11308	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ^*)
87	81	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝐹:𝑍⟶ℝ^*)
88	1	uztrn2 12931	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ 𝑍)
89	88	3adant1 1148	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ 𝑍)
90	87, 89	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ℝ^*)
91	90	ad5ant134 1392	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ^*)
92	53	supxrcld 46004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
93	92	ad5ant13 769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
94		simpr 490	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗))
95	53	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
96		fvres 6900	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ_≥‘𝑖) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) = (𝐹‘𝑗))
97	96	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗))
98	97	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗))
99	3	ffnd 6706	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 Fn 𝑍)
100	99	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹 Fn 𝑍)
101	1, 73	uzssd2 46310	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ 𝑍)
102	101	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ 𝑍)
103		fnssres 6658	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ_≥‘𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
104	100, 102, 103	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
105	104	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
106		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ (ℤ_≥‘𝑖))
107		fnfvelrn 7076	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
108	105, 106, 107	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
109	98, 108	eqeltrd 2860	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
110		eqid 2760	. . . . . . . . . . 11 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )
111	95, 109, 110	supxrubd 46010	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
112	111	ad5ant134 1392	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
113	86, 91, 93, 94, 112	xrletrd 13238	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
114	113	rexlimdva2 3165	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
115	114	ralimdva 3174	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
116	115	reximdva 3175	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
117	84, 116	mpd 16	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
118		simpl 488	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → 𝑦 = 𝑥)
119	76	adantl 487	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
120	118, 119	breq12d 5116	. . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )))
121	120	ralbidva 3183	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) ↔ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )))
122	121	cbvrexvw 3241	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) ↔ ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
123	117, 122	sylibr 237	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ))‘𝑖))
124	1, 2, 33, 79, 123	climinf 46501	. 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) ⇝ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ, < ))
125		fveq2 6881	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑘))
126	125	reseq2d 5974	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑘)))
127	126	rneqd 5924	. . . . . 6 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑘)))
128	127	supeq1d 9423	. . . . 5 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
129	128	cbvmptv 5209	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
130	129	a1i 11	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < )))
131	2, 1, 3, 12	limsupvaluz2 46631	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ))
132	131	eqcomd 2766	. . 3 ⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ) = (lim sup‘𝐹))
133	130, 132	breq12d 5116	. 2 ⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) ⇝ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ, < ) ↔ (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^*, < )) ⇝ (lim sup‘𝐹)))
134	124, 133	mpbid 235	1 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^*, < )) ⇝ (lim sup‘𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3899 ∅c0 4279 class class class wbr 5103 ↦ cmpt 5186 ran crn 5656 ↾ cres 5657 Fn wfn 6530 ⟶wf 6531 ‘cfv 6535 (class class class)co 7416 supcsup 9417 infcinf 9418 ℝcr 11148 1c1 11150 + caddc 11152 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 ℤcz 12640 ℤ_≥cuz 12912 lim supclsp 15582 ⇝ cli 15596

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-er 8703 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9419 df-inf 9420 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-n0 12554 df-z 12641 df-uz 12913 df-rp 13068 df-ico 13429 df-fz 13587 df-fl 13878 df-ceil 13879 df-seq 14091 df-exp 14151 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-limsup 15583 df-clim 15600

This theorem is used by: supcnvlimsupmpt 46634

W3C validator