iunhoiioolem - Mathbox for Glauco Siliprandi

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Theorem iunhoiioolem 45691

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 2731	. . . . . 6 ⊢ (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) = (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
3		iunhoiioolem.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
4		ixpf 8917	. . . . . . . . . . 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 13390	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ⊆ ℝ
7	6	rgenw 3064	. . . . . . . . . . . 12 ⊢ ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
9		iunss 5049	. . . . . . . . . . 11 ⊢ (∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
10	8, 9	sylibr 233	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
11	5, 10	fssd 6736	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
12	11	ffvelcdmda 7087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ)
13		iunhoiioolem.a	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
14	12, 13	resubcld 11647	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
15	13	rexrd 11269	. . . . . . . . 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 44198	. . . . . . . . . 10 ⊢ ((𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ (𝐴(,)𝐵))
20	17, 18, 19	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ (𝐴(,)𝐵))
21		ioogtlb 44508	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐹‘𝑘) ∈ (𝐴(,)𝐵)) → 𝐴 < (𝐹‘𝑘))
22	15, 16, 20, 21	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 < (𝐹‘𝑘))
23	13, 12	posdifd 11806	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴 < (𝐹‘𝑘) ↔ 0 < ((𝐹‘𝑘) − 𝐴)))
24	22, 23	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 < ((𝐹‘𝑘) − 𝐴))
25	14, 24	elrpd 13018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ⁺)
26	1, 2, 25	rnmptssd 44195	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ⁺)
27		iunhoiioolem.c	. . . . . 6 ⊢ 𝐶 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )
28		ltso 11299	. . . . . . . 8 ⊢ < Or ℝ
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → < Or ℝ)
30		iunhoiioolem.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
31	2	rnmptfi 44170	. . . . . . . 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 6244	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ≠ ∅)
35	1, 2, 14	rnmptssd 44195	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)
36		fiinfcl 9499	. . . . . . 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 1371	. . . . . 6 ⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
38	27, 37	eqeltrid 2836	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
39	26, 38	sseldd 3984	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
40		rpgtrecnn 44390	. . . 4 ⊢ (𝐶 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶)
41	39, 40	syl 17	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶)
42	3	elexd 3494	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ V)
43	42	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 ∈ V)
44	5	ffnd 6719	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋)
45	44	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 Fn 𝑋)
46		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
47	1, 46	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
48		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑘(1 / 𝑛)
49		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑘 <
50		nfmpt1 5257	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
51	50	nfrn 5952	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
52		nfcv 2902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘ℝ
53	51, 52, 49	nfinf 9480	. . . . . . . . . . 11 ⊢ Ⅎ𝑘inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )
54	27, 53	nfcxfr 2900	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝐶
55	48, 49, 54	nfbr 5196	. . . . . . . . 9 ⊢ Ⅎ𝑘(1 / 𝑛) < 𝐶
56	47, 55	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶)
57	13	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
58		nnrecre 12259	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
59	58	ad2antlr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
60	57, 59	readdcld 11248	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ)
61	60	rexrd 11269	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ^*)
62	61	adantlr 712	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ^*)
63	16	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
64	63	adantlr 712	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
65		ressxr 11263	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℝ^*
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
67	11, 66	fssd 6736	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ^*)
68	67	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹:𝑋⟶ℝ^*)
69	68	ffvelcdmda 7087	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ^*)
70	60	adantlr 712	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ)
71	12	ad4ant14 749	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ)
72	59	adantlr 712	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
73	35, 38	sseldd 3984	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ ℝ)
74	73	ad3antrrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐶 ∈ ℝ)
75	14	ad4ant14 749	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
76		simplr 766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < 𝐶)
77	35	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)
78	32	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin)
79		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
80		ovexd 7447	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑋 → ((𝐹‘𝑘) − 𝐴) ∈ V)
81	2	elrnmpt1 5958	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐹‘𝑘) − 𝐴) ∈ V) → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
82	79, 80, 81	syl2anc 583	. . . . . . . . . . . . . . . . 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 12205	. . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ≤ ((𝐹‘𝑘) − 𝐴))
86	27, 85	eqbrtrid 5184	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐶 ≤ ((𝐹‘𝑘) − 𝐴))
87	86	adantlr 712	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐶 ≤ ((𝐹‘𝑘) − 𝐴))
88	72, 74, 75, 76, 87	ltletrd 11379	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐹‘𝑘) − 𝐴))
89	57	adantlr 712	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
90	89, 72, 71	ltaddsub2d 11820	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → ((𝐴 + (1 / 𝑛)) < (𝐹‘𝑘) ↔ (1 / 𝑛) < ((𝐹‘𝑘) − 𝐴)))
91	88, 90	mpbird 256	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) < (𝐹‘𝑘))
92	70, 71, 91	ltled 11367	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ≤ (𝐹‘𝑘))
93		iooltub 44523	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐹‘𝑘) ∈ (𝐴(,)𝐵)) → (𝐹‘𝑘) < 𝐵)
94	15, 16, 20, 93	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) < 𝐵)
95	94	ad4ant14 749	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) < 𝐵)
96	62, 64, 69, 92, 95	elicod 13379	. . . . . . . . 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 3253	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵))
99	43, 45, 98	3jca 1127	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
100		elixp2 8898	. . . . . 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 233	. . . . 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 3167	. . 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 5002	. 2 ⊢ (𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) ↔ ∃𝑛 ∈ ℕ 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
106	104, 105	sylibr 233	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 1784 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 Vcvv 3473 ⊆ wss 3949 ∅c0 4323 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 Or wor 5588 ran crn 5678 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 Xcixp 8894 Fincfn 8942 infcinf 9439 ℝcr 11112 0cc0 11113 1c1 11114 + caddc 11116 ℝ^*cxr 11252 < clt 11253 ≤ cle 11254 − cmin 11449 / cdiv 11876 ℕcn 12217 ℝ⁺crp 12979 (,)cioo 13329 [,)cico 13331

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-inf 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-ioo 13333 df-ico 13335 df-fl 13762

This theorem is referenced by: iunhoiioo 45692

W3C validator