Theorem uzub 46441

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 6885	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (ℤ≥‘𝑘) = (ℤ≥‘𝑖))
2	1	raleqdv 3319	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥))
3	2	cbvrexvw 3241	. . . . . 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 5106	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝑤))
6	5	ralbidv 3185	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
7	6	rexbidv 3186	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
8	4, 7	bitrd 282	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
9	8	cbvrexvw 3241	. . 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 5106	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑦))
12	11	ralbidv 3185	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
13	12	rexbidv 3186	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦))
14	13	cbvrexvw 3241	. . . . . 6 ⊢ (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
15	14	biimpi 219	. . . . 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 1946	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑦 ∈ ℝ
18	16, 17	nfan 1931	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑦 ∈ ℝ)
19		nfv 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑖 ∈ 𝑍
20	18, 19	nfan 1931	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍)
21		nfra1 3286	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦
22	20, 21	nfan 1931	. . . . . . . . . 10 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)
23		nfmpt1 5203	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
24	23	nfrn 5933	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵)
25		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗ℝ
26		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 <
27	24, 25, 26	nfsup 9443	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
28		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
29		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦
30	27, 28, 29	nfbr 5151	. . . . . . . . . . 11 ⊢ Ⅎ𝑗sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) ≤ 𝑦
31	30, 29, 27	nfif 4512	. . . . . . . . . 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 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑀 ∈ ℤ)
34		uzub.3	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
35		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → 𝑦 ∈ ℝ)
36		eqid 2760	. . . . . . . . . 10 ⊢ sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < ) = sup(ran (𝑗 ∈ (𝑀...𝑖) ↦ 𝐵), ℝ, < )
37		eqid 2760	. . . . . . . . . 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 781	. . . . . . . . . 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 771	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)
41		simpr 490	. . . . . . . . . 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 46440	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
43	42	rexlimdva2 3165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
44	43	imp 412	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
45	44	rexlimdva2 3165	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
46	45	imp 412	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
47	15, 46	sylan2 605	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤) → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤)
48	47	ex 418	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 → ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
49	32, 34	uzidd2 46426	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
50	49	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → 𝑀 ∈ 𝑍)
51	34	raleqi 3317	. . . . . . 7 ⊢ (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤)
52	51	bilani 510	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤)
53		nfv 1946	. . . . . . 7 ⊢ Ⅎ𝑖∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤
54		fveq2 6885	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (ℤ≥‘𝑖) = (ℤ≥‘𝑀))
55	54	raleqdv 3319	. . . . . . 7 ⊢ (𝑖 = 𝑀 → (∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤))
56	53, 55	rspce 3565	. . . . . 6 ⊢ ((𝑀 ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)𝐵 ≤ 𝑤) → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤)
57	50, 52, 56	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤) → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤)
58	57	ex 418	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 → ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
59	58	reximdva 3175	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 → ∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤))
60	48, 59	impbid 215	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤))
61		breq2 5106	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑥))
62	61	ralbidv 3185	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))
63	62	cbvrexvw 3241	. . 3 ⊢ (∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)
64	63	a1i 11	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑤 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))
65	10, 60, 64	3bitrd 308	1 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ifcif 4481   class class class wbr 5102   ↦ cmpt 5185  ran crn 5651  ‘cfv 6538  (class class class)co 7420  supcsup 9432  ℝcr 11199   < clt 11343   ≤ cle 11344  ℤcz 12693  ℤ_≥cuz 12965  ...cfz 13639

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789

This theorem is used by:  limsupreuz  46747