supcnvlimsup - Mathbox for Glauco Siliprandi

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Theorem supcnvlimsup 43971

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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
5		id 22	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
6	1, 5	uzssd2 43642	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
7	6	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
8	4, 7	feqresmpt 6911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
9	8	rneqd 5893	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)))
10	9	supeq1d 9382	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^, < ))
11		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
12		supcnvlimsup.r	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
13	12	renepnfd 11206	. . . . . . . . 9 ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)
14	11, 1, 3, 13	limsupubuz 43944	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
15	14	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥)
16		ssralv 4010	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥))
19	18	reximdv 3167	. . . . . . 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 1917	. . . . . . 7 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
22	1	eluzelz2 43628	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
23		uzid 12778	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
24		ne0i 4294	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘𝑛) ≠ ∅)
25	22, 23, 24	3syl 18	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
26	25	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ≠ ∅)
27	4	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐹:𝑍⟶ℝ)
28	7	sselda 3944	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
29	27, 28	ffvelcdmd 7036	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ)
30	21, 26, 29	supxrre3rnmpt 43654	. . . . . 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 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ^*, < ) ∈ ℝ)
32	10, 31	eqeltrd 2838	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ) ∈ ℝ)
33	32	fmpttd 7063	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )):𝑍⟶ℝ)
34		eqid 2736	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑖) = (ℤ_≥‘𝑖)
35	1	eluzelz2 43628	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℤ)
36	35	peano2zd 12610	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ ℤ)
37	35	zred 12607	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℝ)
38		lep1 11996	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℝ → 𝑖 ≤ (𝑖 + 1))
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ≤ (𝑖 + 1))
40	34, 35, 36, 39	eluzd 43634	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ (ℤ_≥‘𝑖))
41		uzss 12786	. . . . . . . . 9 ⊢ ((𝑖 + 1) ∈ (ℤ_≥‘𝑖) → (ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖))
42	40, 41	syl 17	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖))
43		ssres2 5965	. . . . . . . 8 ⊢ ((ℤ_≥‘(𝑖 + 1)) ⊆ (ℤ_≥‘𝑖) → (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ_≥‘𝑖)))
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ_≥‘𝑖)))
45		rnss 5894	. . . . . . 7 ⊢ ((𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ_≥‘𝑖)) → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
46	44, 45	syl 17	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
47	46	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
48		rnresss 5973	. . . . . . . 8 ⊢ ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ran 𝐹
49	48	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ran 𝐹)
50	3	frnd 6676	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
51	50	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ)
52	49, 51	sstrd 3954	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ)
53		ressxr 11199	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
54	53	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ℝ ⊆ ℝ^*)
55	52, 54	sstrd 3954	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
56		supxrss 13251	. . . . 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^*, < ))
57	47, 55, 56	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
58		eqidd 2737	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )))
59		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑖 + 1) → (ℤ_≥‘𝑛) = (ℤ_≥‘(𝑖 + 1)))
60	59	reseq2d 5937	. . . . . . . . . 10 ⊢ (𝑛 = (𝑖 + 1) → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))))
61	60	rneqd 5893	. . . . . . . . 9 ⊢ (𝑛 = (𝑖 + 1) → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))))
62	61	supeq1d 9382	. . . . . . . 8 ⊢ (𝑛 = (𝑖 + 1) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
63	62	adantl 482	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = (𝑖 + 1)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
64	1	peano2uzs 12827	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ 𝑍)
65		xrltso 13060	. . . . . . . . 9 ⊢ < Or ℝ^*
66	65	supex 9399	. . . . . . . 8 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^*, < ) ∈ V
67	66	a1i 11	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^*, < ) ∈ V)
68	58, 63, 64, 67	fvmptd 6955	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
69	68	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) = sup(ran (𝐹 ↾ (ℤ_≥‘(𝑖 + 1))), ℝ^, < ))
70		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑖))
71	70	reseq2d 5937	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑖)))
72	71	rneqd 5893	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑖)))
73	72	supeq1d 9382	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
74	73	adantl 482	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = 𝑖) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
75		id 22	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ 𝑍)
76	65	supex 9399	. . . . . . . 8 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ) ∈ V
77	76	a1i 11	. . . . . . 7 ⊢ (𝑖 ∈ 𝑍 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ) ∈ V)
78	58, 74, 75, 77	fvmptd 6955	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
79	78	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
80	69, 79	breq12d 5118	. . . 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^, < )))
81	57, 80	mpbird 256	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘(𝑖 + 1)) ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖))
82		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑗𝐹
83	3	frexr 43609	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
84	82, 2, 1, 83	limsupre3uz 43967	. . . . . . 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)))
85	12, 84	mpbid 231	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥))
86	85	simpld 495	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗))
87		simp-4r 782	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ)
88	87	rexrd 11205	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ^*)
89	83	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝐹:𝑍⟶ℝ^*)
90	1	uztrn2 12782	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ 𝑍)
91	90	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ 𝑍)
92	89, 91	ffvelcdmd 7036	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ℝ^*)
93	92	ad5ant134 1367	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ^*)
94	55	supxrcld 43307	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
95	94	ad5ant13 755	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) ∈ ℝ^)
96		simpr 485	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗))
97	55	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ran (𝐹 ↾ (ℤ_≥‘𝑖)) ⊆ ℝ^*)
98		fvres 6861	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ_≥‘𝑖) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) = (𝐹‘𝑗))
99	98	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗))
100	99	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗))
101	3	ffnd 6669	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 Fn 𝑍)
102	101	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹 Fn 𝑍)
103	1, 75	uzssd2 43642	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ 𝑍)
104	103	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ 𝑍)
105		fnssres 6624	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝑍 ∧ (ℤ_≥‘𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
106	102, 104, 105	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
107	106	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖))
108		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → 𝑗 ∈ (ℤ_≥‘𝑖))
109		fnfvelrn 7031	. . . . . . . . . . . . 13 ⊢ (((𝐹 ↾ (ℤ_≥‘𝑖)) Fn (ℤ_≥‘𝑖) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
110	107, 108, 109	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → ((𝐹 ↾ (ℤ_≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
111	100, 110	eqeltrd 2838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ_≥‘𝑖)))
112		eqid 2736	. . . . . . . . . . 11 ⊢ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )
113	97, 111, 112	supxrubd 43313	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
114	113	ad5ant134 1367	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
115	88, 93, 95, 96, 114	xrletrd 13081	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
116	115	rexlimdva2 3154	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
117	116	ralimdva 3164	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
118	117	reximdva 3165	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ_≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < )))
119	86, 118	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^*, < ))
120		simpl 483	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → 𝑦 = 𝑥)
121	78	adantl 482	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < ))
122	120, 121	breq12d 5118	. . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )))
123	122	ralbidva 3172	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ))‘𝑖) ↔ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ_≥‘𝑖)), ℝ^, < )))
124	123	cbvrexvw 3226	. . . 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^, < ))
125	119, 124	sylibr 233	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < ))‘𝑖))
126	1, 2, 33, 81, 125	climinf 43837	. 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) ⇝ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )), ℝ, < ))
127		fveq2 6842	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑘))
128	127	reseq2d 5937	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ_≥‘𝑛)) = (𝐹 ↾ (ℤ_≥‘𝑘)))
129	128	rneqd 5893	. . . . . 6 ⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ_≥‘𝑛)) = ran (𝐹 ↾ (ℤ_≥‘𝑘)))
130	129	supeq1d 9382	. . . . 5 ⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < ) = sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
131	130	cbvmptv 5218	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < ))
132	131	a1i 11	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^, < )) = (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^, < )))
133	2, 1, 3, 12	limsupvaluz2 43969	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ))
134	133	eqcomd 2742	. . 3 ⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑛)), ℝ^*, < )), ℝ, < ) = (lim sup‘𝐹))
135	132, 134	breq12d 5118	. 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)))
136	126, 135	mpbid 231	1 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ_≥‘𝑘)), ℝ^*, < )) ⇝ (lim sup‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 class class class wbr 5105 ↦ cmpt 5188 ran crn 5634 ↾ cres 5635 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 supcsup 9376 infcinf 9377 ℝcr 11050 1c1 11052 + caddc 11054 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 ℤcz 12499 ℤ_≥cuz 12763 lim supclsp 15352 ⇝ cli 15366

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-ico 13270 df-fz 13425 df-fl 13697 df-ceil 13698 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370

This theorem is referenced by: supcnvlimsupmpt 43972

W3C validator