supcnvlimsup - Mathbox for Glauco Siliprandi

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Theorem supcnvlimsup 45738

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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
5		id 22	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
6	1, 5	uzssd2 45413	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
7	6	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
8	4, 7	feqresmpt 6930	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
9	8	rneqd 5902	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
10	9	supeq1d 9397	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^, < ))
11		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
12		supcnvlimsup.r	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
13	12	renepnfd 11225	. . . . . . . . 9 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
14	11, 1, 3, 13	limsupubuz 45711	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
16		ssralv 4015	. . . . . . . . . 10 ⊢ ((ℤ_≥‘𝑛) ⊆ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
17	6, 16	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
19	18	reximdv 3148	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
20	15, 19	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)
21		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
22	1	eluzelz2 45399	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
23		uzid 12808	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
24		ne0i 4304	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘𝑛) ≠ ∅)
25	22, 23, 24	3syl 18	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
26	25	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ≠ ∅)
27	4	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐹:𝑍⟶ℝ)
28	7	sselda 3946	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
29	27, 28	ffvelcdmd 7057	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
30	21, 26, 29	supxrre3rnmpt 45425	. . . . . 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 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^*, < ) ∈ ℝ)
32	10, 31	eqeltrd 2828	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ) ∈ ℝ)
33	32	fmpttd 7087	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )):𝑍⟶ℝ)
34		eqid 2729	. . . . . . . 8 ⊢ (ℤ_≥‘𝑖) = (ℤ_≥‘𝑖)
35	1	eluzelz2 45399	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℤ)
36	35	peano2zd 12641	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ ℤ)
37	35	zred 12638	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℝ)
38		lep1 12023	. . . . . . . . 9 ⊢ (𝑖 ∈ ℝ → 𝑖 ≤ (𝑖 + 1))
39	37, 38	syl 17	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ≤ (𝑖 + 1))
40	34, 35, 36, 39	eluzd 45405	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ (ℤ_≥‘𝑖))
41		uzss 12816	. . . . . . 7 ⊢ ((𝑖 + 1) ∈ (ℤ_≥‘𝑖) → (ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖))
42		ssres2 5975	. . . . . . 7 ⊢ ((ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖) → (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ_≥‘𝑖)))
43		rnss 5903	. . . . . . 7 ⊢ ((𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ_≥‘𝑖)) → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
44	40, 41, 42, 43	4syl 19	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
45	44	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
46		rnresss 5988	. . . . . . . 8 ⊢ ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ran 𝐹
47	46	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ran 𝐹)
48	3	frnd 6696	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
49	48	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
50	47, 49	sstrd 3957	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ)
51		ressxr 11218	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
52	51	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ℝ ⊆ ℝ^*)
53	50, 52	sstrd 3957	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
54		supxrss 13292	. . . . 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 584	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
56		eqidd 2730	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )))
57		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑖 + 1) → (ℤ_≥‘𝑛) = (ℤ_≥‘(𝑖 + 1)))
58	57	reseq2d 5950	. . . . . . . . . 10 ⊢ (𝑛 = (𝑖 + 1) → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))))
59	58	rneqd 5902	. . . . . . . . 9 ⊢ (𝑛 = (𝑖 + 1) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))))
60	59	supeq1d 9397	. . . . . . . 8 ⊢ (𝑛 = (𝑖 + 1) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
61	60	adantl 481	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = (𝑖 + 1)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
62	1	peano2uzs 12861	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ 𝑍)
63		xrltso 13101	. . . . . . . . 9 ⊢ < Or ℝ^*
64	63	supex 9415	. . . . . . . 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 6975	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
67	66	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
68		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑖))
69	68	reseq2d 5950	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑖)))
70	69	rneqd 5902	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑖)))
71	70	supeq1d 9397	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
72	71	adantl 481	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = 𝑖) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
73		id 22	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ 𝑍)
74	63	supex 9415	. . . . . . . 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 6975	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
77	76	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
78	67, 77	breq12d 5120	. . . 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 257	. . 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 2891	. . . . . . . 8 ⊢ Ⅎ𝑗𝐹
81	3	frexr 45381	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
82	80, 2, 1, 81	limsupre3uz 45734	. . . . . . 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 232	. . . . . 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 494	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗))
85		simp-4r 783	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ)
86	85	rexrd 11224	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ^*)
87	81	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝐹:𝑍⟶ℝ^*)
88	1	uztrn2 12812	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ 𝑍)
89	88	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ 𝑍)
90	87, 89	ffvelcdmd 7057	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ℝ^*)
91	90	ad5ant134 1369	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ^*)
92	53	supxrcld 45101	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
93	92	ad5ant13 756	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
94		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗))
95	53	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
96		fvres 6877	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ_≥‘𝑖) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) = (𝐹‘𝑗))
97	96	eqcomd 2735	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗))
98	97	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗))
99	3	ffnd 6689	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 Fn 𝑍)
100	99	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹 Fn 𝑍)
101	1, 73	uzssd2 45413	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ 𝑍)
102	101	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ 𝑍)
103		fnssres 6641	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ_≥‘𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
104	100, 102, 103	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
105	104	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
106		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ (ℤ_≥‘𝑖))
107		fnfvelrn 7052	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
108	105, 106, 107	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
109	98, 108	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
110		eqid 2729	. . . . . . . . . . 11 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )
111	95, 109, 110	supxrubd 45107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
112	111	ad5ant134 1369	. . . . . . . . 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 13122	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
114	113	rexlimdva2 3136	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
115	114	ralimdva 3145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
116	115	reximdva 3146	. . . . 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 15	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
118		simpl 482	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → 𝑦 = 𝑥)
119	76	adantl 481	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
120	118, 119	breq12d 5120	. . . . . 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 3154	. . . . 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 3216	. . . 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 234	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ))‘𝑖))
124	1, 2, 33, 79, 123	climinf 45604	. 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) ⇝ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ, < ))
125		fveq2 6858	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑘))
126	125	reseq2d 5950	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑘)))
127	126	rneqd 5902	. . . . . 6 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑘)))
128	127	supeq1d 9397	. . . . 5 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
129	128	cbvmptv 5211	. . . 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 45736	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ))
132	131	eqcomd 2735	. . 3 ⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ) = (lim sup‘𝐹))
133	130, 132	breq12d 5120	. 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 232	1 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^*, < )) ⇝ (lim sup‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ⊆ wss 3914 ∅c0 4296 class class class wbr 5107 ↦ cmpt 5188 ran crn 5639 ↾ cres 5640 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 supcsup 9391 infcinf 9392 ℝcr 11067 1c1 11069 + caddc 11071 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 ℤcz 12529 ℤ_≥cuz 12793 lim supclsp 15436 ⇝ cli 15450

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-ico 13312 df-fz 13469 df-fl 13754 df-ceil 13755 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454

This theorem is referenced by: supcnvlimsupmpt 45739

W3C validator