Theorem uzub 45425

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 6881	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (ℤ≥‘𝑘) = (ℤ≥‘𝑖))
2	1	raleqdv 3309	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥))
3	2	cbvrexvw 3225	. . . . . 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 5128	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝑤))
6	5	ralbidv 3164	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
7	6	rexbidv 3165	. . . . 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 3225	. . 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 5128	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
12	11	ralbidv 3164	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
13	12	rexbidv 3165	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
14	13	cbvrexvw 3225	. . . . . 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 3270	. . . . . . . . . . 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 5225	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
24	23	nfrn 5937	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
25		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ℝ
26		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 <
27	24, 25, 26	nfsup 9468	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
28		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
29		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
30	27, 28, 29	nfbr 5171	. . . . . . . . . . 11 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) ≤ 𝑦
31	30, 29, 27	nfif 4536	. . . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) = sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
37		eqid 2736	. . . . . . . . . 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 45424	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
43	42	rexlimdva2 3144	. . . . . . . 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 3144	. . . . . 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 45410	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
50	49	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → 𝑀 ∈ 𝑍)
51	34	raleqi 3307	. . . . . . . 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 6881	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (ℤ≥‘𝑖) = (ℤ≥‘𝑀))
56	55	raleqdv 3309	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤))
57	54, 56	rspce 3595	. . . . . 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 3154	. . 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 5128	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑥))
63	62	ralbidv 3164	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))
64	63	cbvrexvw 3225	. . 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 3052  ∃wrex 3061  ifcif 4505   class class class wbr 5124   ↦ cmpt 5206  ran crn 5660  ‘cfv 6536  (class class class)co 7410  supcsup 9457  ℝcr 11133   < clt 11274   ≤ cle 11275  ℤcz 12593  ℤ_≥cuz 12857  ...cfz 13529

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  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 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530  df-fzo 13677

This theorem is referenced by:  limsupreuz  45733