Theorem uzub 45712

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 6833	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (ℤ≥‘𝑘) = (ℤ≥‘𝑖))
2	1	raleqdv 3295	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥))
3	2	cbvrexvw 3214	. . . . . 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 5101	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝑤))
6	5	ralbidv 3158	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
7	6	rexbidv 3159	. . . . 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 3214	. . 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 5101	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
12	11	ralbidv 3158	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
13	12	rexbidv 3159	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
14	13	cbvrexvw 3214	. . . . . 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 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑦 ∈ ℝ
18	16, 17	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ)
19		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 ∈ 𝑍
20	18, 19	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍)
21		nfra1 3259	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦
22	20, 21	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
23		nfmpt1 5196	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
24	23	nfrn 5900	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
25		nfcv 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ℝ
26		nfcv 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 <
27	24, 25, 26	nfsup 9356	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
28		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
29		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
30	27, 28, 29	nfbr 5144	. . . . . . . . . . 11 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) ≤ 𝑦
31	30, 29, 27	nfif 4509	. . . . . . . . . 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 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑀 ∈ ℤ)
34		uzub.3	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
35		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑦 ∈ ℝ)
36		eqid 2735	. . . . . . . . . 10 ⊢ sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) = sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
37		eqid 2735	. . . . . . . . . 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 769	. . . . . . . . . 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 759	. . . . . . . . . 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 45711	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
43	42	rexlimdva2 3138	. . . . . . . 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 3138	. . . . . 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 594	. . . 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 45697	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
50	49	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → 𝑀 ∈ 𝑍)
51	34	raleqi 3293	. . . . . . . 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 1916	. . . . . . 7 ⊢ Ⅎ𝑖∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤
55		fveq2 6833	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (ℤ≥‘𝑖) = (ℤ≥‘𝑀))
56	55	raleqdv 3295	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤))
57	54, 56	rspce 3564	. . . . . 6 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤) → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤)
58	50, 53, 57	syl2anc 585	. . . . 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 3148	. . 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 5101	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑥))
63	62	ralbidv 3158	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))
64	63	cbvrexvw 3214	. . 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 1542  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3050  ∃wrex 3059  ifcif 4478   class class class wbr 5097   ↦ cmpt 5178  ran crn 5624  ‘cfv 6491  (class class class)co 7358  supcsup 9345  ℝcr 11027   < clt 11168   ≤ cle 11169  ℤcz 12490  ℤ_≥cuz 12753  ...cfz 13425

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-sup 9347  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-n0 12404  df-z 12491  df-uz 12754  df-fz 13426  df-fzo 13573

This theorem is referenced by:  limsupreuz  46018