Theorem uzub 44842

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 6902	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (ℤ≥‘𝑘) = (ℤ≥‘𝑖))
2	1	raleqdv 3323	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥))
3	2	cbvrexvw 3233	. . . . . 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 5156	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝑤))
6	5	ralbidv 3175	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
7	6	rexbidv 3176	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
8	4, 7	bitrd 278	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
9	8	cbvrexvw 3233	. . 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 5156	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
12	11	ralbidv 3175	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
13	12	rexbidv 3176	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
14	13	cbvrexvw 3233	. . . . . 6 ⊢ (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
15	14	biimpi 215	. . . . 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 1909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑦 ∈ ℝ
18	16, 17	nfan 1894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ)
19		nfv 1909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 ∈ 𝑍
20	18, 19	nfan 1894	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍)
21		nfra1 3279	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦
22	20, 21	nfan 1894	. . . . . . . . . 10 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
23		nfmpt1 5260	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
24	23	nfrn 5958	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
25		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ℝ
26		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 <
27	24, 25, 26	nfsup 9482	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
28		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
29		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
30	27, 28, 29	nfbr 5199	. . . . . . . . . . 11 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) ≤ 𝑦
31	30, 29, 27	nfif 4562	. . . . . . . . . 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 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑀 ∈ ℤ)
34		uzub.3	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
35		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑦 ∈ ℝ)
36		eqid 2728	. . . . . . . . . 10 ⊢ sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) = sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
37		eqid 2728	. . . . . . . . . 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 767	. . . . . . . . . 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 757	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)
41		simpr 483	. . . . . . . . . 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 44841	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
43	42	rexlimdva2 3154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
44	43	imp 405	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
45	44	rexlimdva2 3154	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
46	45	imp 405	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
47	15, 46	sylan2 591	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
48	47	ex 411	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
49	32, 34	uzidd2 44827	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
50	49	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → 𝑀 ∈ 𝑍)
51	34	raleqi 3321	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤)
52	51	biimpi 215	. . . . . . 7 ⊢ (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 → ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤)
53	52	adantl 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤)
54		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑖∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤
55		fveq2 6902	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (ℤ≥‘𝑖) = (ℤ≥‘𝑀))
56	55	raleqdv 3323	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤))
57	54, 56	rspce 3600	. . . . . 6 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤) → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤)
58	50, 53, 57	syl2anc 582	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤)
59	58	ex 411	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
60	59	reximdva 3165	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 → ∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
61	48, 60	impbid 211	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
62		breq2 5156	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑥))
63	62	ralbidv 3175	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))
64	63	cbvrexvw 3233	. . 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 304	1 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067  ifcif 4532   class class class wbr 5152   ↦ cmpt 5235  ran crn 5683  ‘cfv 6553  (class class class)co 7426  supcsup 9471  ℝcr 11145   < clt 11286   ≤ cle 11287  ℤcz 12596  ℤ_≥cuz 12860  ...cfz 13524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668

This theorem is referenced by:  limsupreuz  45154