iunhoiioolem - Mathbox for Glauco Siliprandi

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Theorem iunhoiioolem 47254

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 2764	. . . . . 6 ⊢ (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) = (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
3		iunhoiioolem.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
4		ixpf 8904	. . . . . . . . . . 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 13413	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ⊆ ℝ
7	6	rgenw 3082	. . . . . . . . . . . 12 ⊢ ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
9		iunss 5004	. . . . . . . . . . 11 ⊢ (∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
10	8, 9	sylibr 236	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
11	5, 10	fssd 6711	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
12	11	ffvelcdmda 7067	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ)
13		iunhoiioolem.a	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
14	12, 13	resubcld 11617	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
15	13	rexrd 11234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ^*)
16		iunhoiioolem.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
17	3	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
18		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
19		fvixp2 45781	. . . . . . . . . 10 ⊢ ((𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ (𝐴(,)𝐵))
20	17, 18, 19	syl2anc 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ (𝐴(,)𝐵))
21		ioogtlb 46076	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐹‘𝑘) ∈ (𝐴(,)𝐵)) → 𝐴 < (𝐹‘𝑘))
22	15, 16, 20, 21	syl3anc 1392	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 < (𝐹‘𝑘))
23	13, 12	posdifd 11776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴 < (𝐹‘𝑘) ↔ 0 < ((𝐹‘𝑘) − 𝐴)))
24	22, 23	mpbid 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 < ((𝐹‘𝑘) − 𝐴))
25	14, 24	elrpd 13036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ⁺)
26	1, 2, 25	rnmptssd 7107	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ⁺)
27		iunhoiioolem.c	. . . . . 6 ⊢ 𝐶 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )
28		ltso 11265	. . . . . . . 8 ⊢ < Or ℝ
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → < Or ℝ)
30		iunhoiioolem.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
31	2	rnmptfi 45754	. . . . . . . 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 7107	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)
36		fiinfcl 9451	. . . . . . 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 1393	. . . . . 6 ⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
38	27, 37	eqeltrid 2868	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
39	26, 38	sseldd 3939	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
40		rpgtrecnn 45960	. . . 4 ⊢ (𝐶 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶)
41	39, 40	syl 17	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶)
42	3	elexd 3479	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ V)
43	42	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 ∈ V)
44	5	ffnd 6694	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋)
45	44	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 Fn 𝑋)
46		nfv 1936	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
47	1, 46	nfan 1921	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
48		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑘(1 / 𝑛)
49		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑘 <
50		nfmpt1 5201	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
51	50	nfrn 5930	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
52		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘ℝ
53	51, 52, 49	nfinf 9431	. . . . . . . . . . 11 ⊢ Ⅎ𝑘inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )
54	27, 53	nfcxfr 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝐶
55	48, 49, 54	nfbr 5149	. . . . . . . . 9 ⊢ Ⅎ𝑘(1 / 𝑛) < 𝐶
56	47, 55	nfan 1921	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶)
57	13	adantlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
58		nnrecre 12257	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
59	58	ad2antlr 737	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
60	57, 59	readdcld 11213	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ)
61	60	rexrd 11234	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ^*)
62	61	adantlr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ^*)
63	16	adantlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
64	63	adantlr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
65		ressxr 11228	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℝ^*
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
67	11, 66	fssd 6711	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ^*)
68	67	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹:𝑋⟶ℝ^*)
69	68	ffvelcdmda 7067	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ^*)
70	60	adantlr 725	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ)
71	12	ad4ant14 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ)
72	59	adantlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
73	35, 38	sseldd 3939	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ ℝ)
74	73	ad3antrrr 740	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐶 ∈ ℝ)
75	14	ad4ant14 762	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
76		simplr 778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < 𝐶)
77	35	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)
78	32	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin)
79		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
80		ovexd 7433	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑋 → ((𝐹‘𝑘) − 𝐴) ∈ V)
81	2	elrnmpt1 5938	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐹‘𝑘) − 𝐴) ∈ V) → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
82	79, 80, 81	syl2anc 593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
83	82	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
84		infrefilb 12180	. . . . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ≤ ((𝐹‘𝑘) − 𝐴))
86	27, 85	eqbrtrid 5137	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐶 ≤ ((𝐹‘𝑘) − 𝐴))
87	86	adantlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐶 ≤ ((𝐹‘𝑘) − 𝐴))
88	72, 74, 75, 76, 87	ltletrd 11345	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐹‘𝑘) − 𝐴))
89	57	adantlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
90	89, 72, 71	ltaddsub2d 11790	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → ((𝐴 + (1 / 𝑛)) < (𝐹‘𝑘) ↔ (1 / 𝑛) < ((𝐹‘𝑘) − 𝐴)))
91	88, 90	mpbird 259	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) < (𝐹‘𝑘))
92	70, 71, 91	ltled 11333	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ≤ (𝐹‘𝑘))
93		iooltub 46091	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐹‘𝑘) ∈ (𝐴(,)𝐵)) → (𝐹‘𝑘) < 𝐵)
94	15, 16, 20, 93	syl3anc 1392	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) < 𝐵)
95	94	ad4ant14 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) < 𝐵)
96	62, 64, 69, 92, 95	elicod 13401	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵))
97	96	ex 416	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → (𝑘 ∈ 𝑋 → (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
98	56, 97	ralrimi 3262	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵))
99	43, 45, 98	3jca 1142	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
100		elixp2 8885	. . . . . 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 236	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
102	101	ex 416	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) < 𝐶 → 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵)))
103	102	reximdva 3177	. . 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 4955	. 2 ⊢ (𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) ↔ ∃𝑛 ∈ ℕ 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
106	104, 105	sylibr 236	1 ⊢ (𝜑 → 𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1099 = wceq 1562 Ⅎwnf 1805 ∈ wcel 2144 ≠ wne 2959 ∀wral 3078 ∃wrex 3088 Vcvv 3456 ⊆ wss 3906 ∅c0 4287 ∪ ciun 4951 class class class wbr 5102 ↦ cmpt 5183 Or wor 5556 ran crn 5650 Fn wfn 6518 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 Xcixp 8881 Fincfn 8929 infcinf 9389 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 − cmin 11416 / cdiv 11846 ℕcn 12212 ℝ⁺crp 12995 (,)cioo 13351 [,)cico 13353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8680 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-sup 9390 df-inf 9391 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-n0 12484 df-z 12571 df-uz 12842 df-rp 12996 df-ioo 13355 df-ico 13357 df-fl 13804

This theorem is referenced by: iunhoiioo 47255

W3C validator