iunhoiioolem - Mathbox for Glauco Siliprandi

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Theorem iunhoiioolem 46704

Description: A n-dimensional open interval expressed as the indexed union of half-open intervals. One side of the double inclusion. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
iunhoiioolem.K	⊢ Ⅎ𝑘𝜑
iunhoiioolem.x	⊢ (𝜑 → 𝑋 ∈ Fin)
iunhoiioolem.n	⊢ (𝜑 → 𝑋 ≠ ∅)
iunhoiioolem.a	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
iunhoiioolem.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
iunhoiioolem.f	⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
iunhoiioolem.c	⊢ 𝐶 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )

**Assertion**
Ref	Expression
iunhoiioolem	⊢ (𝜑 → 𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))

Distinct variable groups: 𝐶,𝑛 𝑘,𝐹,𝑛 𝑘,𝑋 𝜑,𝑛 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘,𝑛) 𝐵(𝑘,𝑛) 𝐶(𝑘) 𝑋(𝑛)

**Proof of Theorem iunhoiioolem**
Step	Hyp	Ref	Expression
1		iunhoiioolem.K	. . . . . 6 ⊢ Ⅎ𝑘𝜑
2		eqid 2735	. . . . . 6 ⊢ (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) = (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
3		iunhoiioolem.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
4		ixpf 8934	. . . . . . . . . . 11 ⊢ (𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵) → 𝐹:𝑋⟶∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵))
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵))
6		ioossre 13424	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ⊆ ℝ
7	6	rgenw 3055	. . . . . . . . . . . 12 ⊢ ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
9		iunss 5021	. . . . . . . . . . 11 ⊢ (∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
10	8, 9	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
11	5, 10	fssd 6723	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
12	11	ffvelcdmda 7074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ)
13		iunhoiioolem.a	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
14	12, 13	resubcld 11665	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
15	13	rexrd 11285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ^*)
16		iunhoiioolem.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
17	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
18		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
19		fvixp2 45223	. . . . . . . . . 10 ⊢ ((𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ (𝐴(,)𝐵))
20	17, 18, 19	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ (𝐴(,)𝐵))
21		ioogtlb 45524	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐹‘𝑘) ∈ (𝐴(,)𝐵)) → 𝐴 < (𝐹‘𝑘))
22	15, 16, 20, 21	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 < (𝐹‘𝑘))
23	13, 12	posdifd 11824	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴 < (𝐹‘𝑘) ↔ 0 < ((𝐹‘𝑘) − 𝐴)))
24	22, 23	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 < ((𝐹‘𝑘) − 𝐴))
25	14, 24	elrpd 13048	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ⁺)
26	1, 2, 25	rnmptssd 45220	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ⁺)
27		iunhoiioolem.c	. . . . . 6 ⊢ 𝐶 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )
28		ltso 11315	. . . . . . . 8 ⊢ < Or ℝ
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → < Or ℝ)
30		iunhoiioolem.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
31	2	rnmptfi 45195	. . . . . . . 8 ⊢ (𝑋 ∈ Fin → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin)
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin)
33		iunhoiioolem.n	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ ∅)
34	1, 14, 2, 33	rnmptn0 6233	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ≠ ∅)
35	1, 2, 14	rnmptssd 45220	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)
36		fiinfcl 9515	. . . . . . 7 ⊢ (( < Or ℝ ∧ (ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ≠ ∅ ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
37	29, 32, 34, 35, 36	syl13anc 1374	. . . . . 6 ⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
38	27, 37	eqeltrid 2838	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
39	26, 38	sseldd 3959	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
40		rpgtrecnn 45407	. . . 4 ⊢ (𝐶 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶)
41	39, 40	syl 17	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶)
42	3	elexd 3483	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ V)
43	42	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 ∈ V)
44	5	ffnd 6707	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋)
45	44	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 Fn 𝑋)
46		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
47	1, 46	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
48		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑘(1 / 𝑛)
49		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑘 <
50		nfmpt1 5220	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
51	50	nfrn 5932	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
52		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘ℝ
53	51, 52, 49	nfinf 9495	. . . . . . . . . . 11 ⊢ Ⅎ𝑘inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )
54	27, 53	nfcxfr 2896	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝐶
55	48, 49, 54	nfbr 5166	. . . . . . . . 9 ⊢ Ⅎ𝑘(1 / 𝑛) < 𝐶
56	47, 55	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶)
57	13	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
58		nnrecre 12282	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
59	58	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
60	57, 59	readdcld 11264	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ)
61	60	rexrd 11285	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ^*)
62	61	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ^*)
63	16	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
64	63	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
65		ressxr 11279	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℝ^*
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
67	11, 66	fssd 6723	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ^*)
68	67	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹:𝑋⟶ℝ^*)
69	68	ffvelcdmda 7074	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ^*)
70	60	adantlr 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ)
71	12	ad4ant14 752	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ)
72	59	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
73	35, 38	sseldd 3959	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ ℝ)
74	73	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐶 ∈ ℝ)
75	14	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
76		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < 𝐶)
77	35	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)
78	32	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin)
79		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
80		ovexd 7440	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑋 → ((𝐹‘𝑘) − 𝐴) ∈ V)
81	2	elrnmpt1 5940	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐹‘𝑘) − 𝐴) ∈ V) → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
82	79, 80, 81	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
83	82	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
84		infrefilb 12228	. . . . . . . . . . . . . . . 16 ⊢ ((ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin ∧ ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ≤ ((𝐹‘𝑘) − 𝐴))
85	77, 78, 83, 84	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ≤ ((𝐹‘𝑘) − 𝐴))
86	27, 85	eqbrtrid 5154	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐶 ≤ ((𝐹‘𝑘) − 𝐴))
87	86	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐶 ≤ ((𝐹‘𝑘) − 𝐴))
88	72, 74, 75, 76, 87	ltletrd 11395	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐹‘𝑘) − 𝐴))
89	57	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
90	89, 72, 71	ltaddsub2d 11838	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → ((𝐴 + (1 / 𝑛)) < (𝐹‘𝑘) ↔ (1 / 𝑛) < ((𝐹‘𝑘) − 𝐴)))
91	88, 90	mpbird 257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) < (𝐹‘𝑘))
92	70, 71, 91	ltled 11383	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ≤ (𝐹‘𝑘))
93		iooltub 45539	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐹‘𝑘) ∈ (𝐴(,)𝐵)) → (𝐹‘𝑘) < 𝐵)
94	15, 16, 20, 93	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) < 𝐵)
95	94	ad4ant14 752	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) < 𝐵)
96	62, 64, 69, 92, 95	elicod 13412	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵))
97	96	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → (𝑘 ∈ 𝑋 → (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
98	56, 97	ralrimi 3240	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵))
99	43, 45, 98	3jca 1128	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
100		elixp2 8915	. . . . . 6 ⊢ (𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
101	99, 100	sylibr 234	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
102	101	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) < 𝐶 → 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵)))
103	102	reximdva 3153	. . 3 ⊢ (𝜑 → (∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶 → ∃𝑛 ∈ ℕ 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵)))
104	41, 103	mpd 15	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
105		eliun 4971	. 2 ⊢ (𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) ↔ ∃𝑛 ∈ ℕ 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
106	104, 105	sylibr 234	1 ⊢ (𝜑 → 𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 Vcvv 3459 ⊆ wss 3926 ∅c0 4308 ∪ ciun 4967 class class class wbr 5119 ↦ cmpt 5201 Or wor 5560 ran crn 5655 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 Xcixp 8911 Fincfn 8959 infcinf 9453 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 − cmin 11466 / cdiv 11894 ℕcn 12240 ℝ⁺crp 13008 (,)cioo 13362 [,)cico 13364

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-ioo 13366 df-ico 13368 df-fl 13809

This theorem is referenced by: iunhoiioo 46705

W3C validator