supcnvlimsup - Mathbox for Glauco Siliprandi

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Theorem supcnvlimsup 46482

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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
5		id 23	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
6	1, 5	uzssd2 46159	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
7	6	adantl 486	. . . . . . . 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 5928	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
10	9	supeq1d 9402	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^, < ))
11		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
12		supcnvlimsup.r	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
13	12	renepnfd 11264	. . . . . . . . 9 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
14	11, 1, 3, 13	limsupubuz 46455	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
15	14	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
16		ssralv 4006	. . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
19	18	reximdv 3180	. . . . . . 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 1944	. . . . . . 7 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
22	1	eluzelz2 46145	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
23		uzid 12881	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
24		ne0i 4294	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘𝑛) ≠ ∅)
25	22, 23, 24	3syl 19	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
26	25	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ≠ ∅)
27	4	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐹:𝑍⟶ℝ)
28	7	sselda 3937	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
29	27, 28	ffvelcdmd 7080	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
30	21, 26, 29	supxrre3rnmpt 46171	. . . . . 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 2863	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ) ∈ ℝ)
33	32	fmpttd 7110	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )):𝑍⟶ℝ)
34		eqid 2763	. . . . . . . 8 ⊢ (ℤ_≥‘𝑖) = (ℤ_≥‘𝑖)
35	1	eluzelz2 46145	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℤ)
36	35	peano2zd 12707	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ ℤ)
37	35	zred 12704	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℝ)
38		lep1 12060	. . . . . . . . 9 ⊢ (𝑖 ∈ ℝ → 𝑖 ≤ (𝑖 + 1))
39	37, 38	syl 18	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ≤ (𝑖 + 1))
40	34, 35, 36, 39	eluzd 46151	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ (ℤ_≥‘𝑖))
41		uzss 12889	. . . . . . 7 ⊢ ((𝑖 + 1) ∈ (ℤ_≥‘𝑖) → (ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖))
42		ssres2 6003	. . . . . . 7 ⊢ ((ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖) → (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ_≥‘𝑖)))
43		rnss 5929	. . . . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
46		rnresss 6016	. . . . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
50	47, 49	sstrd 3947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ)
51		ressxr 11257	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
52	51	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ℝ ⊆ ℝ^*)
53	50, 52	sstrd 3947	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
54		supxrss 13362	. . . . 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 595	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
56		eqidd 2764	. . . . . . 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 5978	. . . . . . . . . 10 ⊢ (𝑛 = (𝑖 + 1) → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))))
59	58	rneqd 5928	. . . . . . . . 9 ⊢ (𝑛 = (𝑖 + 1) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))))
60	59	supeq1d 9402	. . . . . . . 8 ⊢ (𝑛 = (𝑖 + 1) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
61	60	adantl 486	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = (𝑖 + 1)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
62	1	peano2uzs 12930	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ 𝑍)
63		xrltso 13170	. . . . . . . . 9 ⊢ < Or ℝ^*
64	63	supex 9420	. . . . . . . 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 486	. . . . 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 5978	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑖)))
70	69	rneqd 5928	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑖)))
71	70	supeq1d 9402	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
72	71	adantl 486	. . . . . . 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 9420	. . . . . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
78	67, 77	breq12d 5122	. . . 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 2925	. . . . . . . 8 ⊢ Ⅎ𝑗𝐹
81	3	frexr 46128	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
82	80, 2, 1, 81	limsupre3uz 46478	. . . . . . 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 499	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗))
85		simp-4r 795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ)
86	85	rexrd 11263	. . . . . . . . 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 12885	. . . . . . . . . . . 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 7080	. . . . . . . . . 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 45853	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
93	92	ad5ant13 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
94		simpr 489	. . . . . . . . 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 2769	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹 Fn 𝑍)
101	1, 73	uzssd2 46159	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ 𝑍)
102	101	adantl 486	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 7075	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
108	105, 106, 107	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
109	98, 108	eqeltrd 2863	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
110		eqid 2763	. . . . . . . . . . 11 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )
111	95, 109, 110	supxrubd 45859	. . . . . . . . . 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 13191	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
114	113	rexlimdva2 3168	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
115	114	ralimdva 3177	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
116	115	reximdva 3178	. . . . 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 487	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → 𝑦 = 𝑥)
119	76	adantl 486	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
120	118, 119	breq12d 5122	. . . . . 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 3186	. . . . 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 3244	. . . 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 46350	. 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 5978	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑘)))
127	126	rneqd 5928	. . . . . 6 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑘)))
128	127	supeq1d 9402	. . . . 5 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
129	128	cbvmptv 5215	. . . 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 46480	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ))
132	131	eqcomd 2769	. . 3 ⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ) = (lim sup‘𝐹))
133	130, 132	breq12d 5122	. 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 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ⊆ wss 3905 ∅c0 4286 class class class wbr 5109 ↦ cmpt 5192 ran crn 5662 ↾ cres 5663 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 supcsup 9396 infcinf 9397 ℝcr 11103 1c1 11105 + caddc 11107 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 ℤcz 12595 ℤ_≥cuz 12866 lim supclsp 15526 ⇝ cli 15540

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-n0 12509 df-z 12596 df-uz 12867 df-rp 13021 df-ico 13382 df-fz 13540 df-fl 13830 df-ceil 13831 df-seq 14043 df-exp 14103 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-limsup 15527 df-clim 15544

This theorem is used by: supcnvlimsupmpt 46483

W3C validator