Theorem uzub 45430

Description: A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
uzub.1	⊢ Ⅎ𝑗𝜑
uzub.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
uzub.3	⊢ 𝑍 = (ℤ≥‘𝑀)
uzub.12	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
uzub	⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))

Distinct variable groups:   𝐵,𝑘,𝑥   𝑗,𝑀   𝑗,𝑍,𝑘,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐵(𝑗)   𝑀(𝑥,𝑘)

Proof of Theorem uzub

Dummy variables 𝑖 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (ℤ≥‘𝑘) = (ℤ≥‘𝑖))
2	1	raleqdv 3289	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥))
3	2	cbvrexvw 3208	. . . . . 6 ⊢ (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥)
4	3	a1i 11	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥))
5		breq2 5096	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝑤))
6	5	ralbidv 3152	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
7	6	rexbidv 3153	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
8	4, 7	bitrd 279	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
9	8	cbvrexvw 3208	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤)
10	9	a1i 11	. 2 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
11		breq2 5096	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
12	11	ralbidv 3152	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
13	12	rexbidv 3153	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
14	13	cbvrexvw 3208	. . . . . 6 ⊢ (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
15	14	biimpi 216	. . . . 5 ⊢ (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
16		uzub.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
17		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑦 ∈ ℝ
18	16, 17	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ)
19		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 ∈ 𝑍
20	18, 19	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍)
21		nfra1 3253	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦
22	20, 21	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
23		nfmpt1 5191	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
24	23	nfrn 5894	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
25		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ℝ
26		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 <
27	24, 25, 26	nfsup 9341	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
28		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
29		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
30	27, 28, 29	nfbr 5139	. . . . . . . . . . 11 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) ≤ 𝑦
31	30, 29, 27	nfif 4507	. . . . . . . . . 10 ⊢ Ⅎ𝑗if(sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ))
32		uzub.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
33	32	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑀 ∈ ℤ)
34		uzub.3	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
35		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑦 ∈ ℝ)
36		eqid 2729	. . . . . . . . . 10 ⊢ sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) = sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
37		eqid 2729	. . . . . . . . . 10 ⊢ if(sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )) = if(sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ))
38		simplr 768	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑖 ∈ 𝑍)
39		uzub.12	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)
40	39	ad5ant15 758	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)
41		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
42	22, 31, 33, 34, 35, 36, 37, 38, 40, 41	uzublem 45429	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
43	42	rexlimdva2 3132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
44	43	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
45	44	rexlimdva2 3132	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
46	45	imp 406	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
47	15, 46	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
48	47	ex 412	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
49	32, 34	uzidd2 45415	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
50	49	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → 𝑀 ∈ 𝑍)
51	34	raleqi 3287	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤)
52	51	biimpi 216	. . . . . . 7 ⊢ (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 → ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤)
53	52	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤)
54		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑖∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤
55		fveq2 6822	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (ℤ≥‘𝑖) = (ℤ≥‘𝑀))
56	55	raleqdv 3289	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤))
57	54, 56	rspce 3566	. . . . . 6 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤) → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤)
58	50, 53, 57	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤)
59	58	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
60	59	reximdva 3142	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 → ∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
61	48, 60	impbid 212	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
62		breq2 5096	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑥))
63	62	ralbidv 3152	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))
64	63	cbvrexvw 3208	. . 3 ⊢ (∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)
65	64	a1i 11	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))
66	10, 61, 65	3bitrd 305	1 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ifcif 4476   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620  ‘cfv 6482  (class class class)co 7349  supcsup 9330  ℝcr 11008   < clt 11149   ≤ cle 11150  ℤcz 12471  ℤ_≥cuz 12735  ...cfz 13410

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-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558

This theorem is referenced by:  limsupreuz  45738