qndenserrnbllem - Mathbox for Glauco Siliprandi

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Theorem qndenserrnbllem 42936

Description: n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
qndenserrnbllem.i	⊢ (𝜑 → 𝐼 ∈ Fin)
qndenserrnbllem.n	⊢ (𝜑 → 𝐼 ≠ ∅)
qndenserrnbllem.x	⊢ (𝜑 → 𝑋 ∈ (ℝ ↑_m 𝐼))
qndenserrnbllem.d	⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
qndenserrnbllem.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)

**Assertion**
Ref	Expression
qndenserrnbllem	⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Distinct variable groups: 𝑦,𝐸 𝑦,𝐼 𝑦,𝑋 𝜑,𝑦 Allowed substitution hint: 𝐷(𝑦)

Proof of Theorem qndenserrnbllem Dummy variables 𝑖 𝑘 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qndenserrnbllem.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
2		inss1 4155	. . . . . 6 ⊢ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ
3		qex 12348	. . . . . 6 ⊢ ℚ ∈ V
4		ssexg 5191	. . . . . 6 ⊢ (((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ ∧ ℚ ∈ V) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∈ V)
5	2, 3, 4	mp2an 691	. . . . 5 ⊢ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∈ V
6	5	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∈ V)
7		qndenserrnbllem.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑_m 𝐼))
8		elmapi 8411	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → 𝑋:𝐼⟶ℝ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋:𝐼⟶ℝ)
10	9	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑋:𝐼⟶ℝ)
11		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
12	10, 11	ffvelrnd 6829	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ)
13	12	rexrd 10680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ^*)
14		qndenserrnbllem.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
15	14	rpred 12419	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ)
16	15	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ)
17		ne0i 4250	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐼 → 𝐼 ≠ ∅)
18	17	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ≠ ∅)
19		hashnncl 13723	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ Fin → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
20	1, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
21	20	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
22	18, 21	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℕ)
23	22	nnred 11640	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ)
24		0red 10633	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ∈ ℝ)
25	22	nngt0d 11674	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (♯‘𝐼))
26	24, 23, 25	ltled 10777	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ≤ (♯‘𝐼))
27	23, 26	resqrtcld 14769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ)
28	23, 25	elrpd 12416	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ⁺)
29	28	sqrtgt0d 14764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (√‘(♯‘𝐼)))
30	24, 29	gtned 10764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ≠ 0)
31	16, 27, 30	redivcld 11457	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
32	12, 31	readdcld 10659	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
33	32	rexrd 10680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
34	14	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ⁺)
35	27, 29	elrpd 12416	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
36	34, 35	rpdivcld 12436	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ⁺)
37	12, 36	ltaddrpd 12452	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))
38		qbtwnxr 12581	. . . . . . . 8 ⊢ (((𝑋‘𝑘) ∈ ℝ^* ∧ ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^* ∧ (𝑋‘𝑘) < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))) → ∃𝑞 ∈ ℚ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))
39	13, 33, 37, 38	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞 ∈ ℚ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))
40		df-rex 3112	. . . . . . 7 ⊢ (∃𝑞 ∈ ℚ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))) ↔ ∃𝑞(𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
41	39, 40	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞(𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
42		simprl 770	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ ℚ)
43	13	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑋‘𝑘) ∈ ℝ^*)
44	33	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
45		qre 12341	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ℚ → 𝑞 ∈ ℝ)
46	45	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ ℝ)
47		simprrl 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑋‘𝑘) < 𝑞)
48		simprrr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))
49	43, 44, 46, 47, 48	eliood 42135	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))
50	42, 49	elind 4121	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
51	50	ex 416	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))))
52	51	eximdv 1918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∃𝑞(𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → ∃𝑞 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))))
53	41, 52	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
54		n0 4260	. . . . 5 ⊢ ((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ≠ ∅ ↔ ∃𝑞 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
55	53, 54	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ≠ ∅)
56	1, 6, 55	choicefi 41829	. . 3 ⊢ (𝜑 → ∃𝑦(𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))))
57	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 Fn 𝐼 → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ)
58	57	sseld 3914	. . . . . . . . . . 11 ⊢ (𝑦 Fn 𝐼 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑦‘𝑘) ∈ ℚ))
59	58	ralimdv 3145	. . . . . . . . . 10 ⊢ (𝑦 Fn 𝐼 → (∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
60	59	imdistani 572	. . . . . . . . 9 ⊢ ((𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
61		ffnfv 6859	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℚ ↔ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
62	60, 61	sylibr 237	. . . . . . . 8 ⊢ ((𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑦:𝐼⟶ℚ)
63	62	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦:𝐼⟶ℚ)
64	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℚ ∈ V)
65		elmapg 8402	. . . . . . . . 9 ⊢ ((ℚ ∈ V ∧ 𝐼 ∈ Fin) → (𝑦 ∈ (ℚ ↑_m 𝐼) ↔ 𝑦:𝐼⟶ℚ))
66	64, 1, 65	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (ℚ ↑_m 𝐼) ↔ 𝑦:𝐼⟶ℚ))
67	66	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑦 ∈ (ℚ ↑_m 𝐼) ↔ 𝑦:𝐼⟶ℚ))
68	63, 67	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦 ∈ (ℚ ↑_m 𝐼))
69		reex 10617	. . . . . . . . . . 11 ⊢ ℝ ∈ V
70	45	ssriv 3919	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
71		mapss 8436	. . . . . . . . . . 11 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼))
72	69, 70, 71	mp2an 691	. . . . . . . . . 10 ⊢ (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼)
73	72	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼))
74	73, 68	sseldd 3916	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦 ∈ (ℝ ↑_m 𝐼))
75	1	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐼 ∈ Fin)
76		qndenserrnbllem.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ≠ ∅)
77	76	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐼 ≠ ∅)
78		eqid 2798	. . . . . . . . . 10 ⊢ (♯‘𝐼) = (♯‘𝐼)
79	7	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑋 ∈ (ℝ ↑_m 𝐼))
80		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝜑)
81		fveq2 6645	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑦‘𝑘) = (𝑦‘𝑖))
82		fveq2 6645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → (𝑋‘𝑘) = (𝑋‘𝑖))
83	82	oveq1d 7150	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) = ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
84	82, 83	oveq12d 7153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))) = ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
85	84	ineq2d 4139	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) = (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
86	81, 85	eleq12d 2884	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ↔ (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))))
87	86	cbvralvw 3396	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ↔ ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
88	87	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
89	88	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
90		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
91		rspa 3171	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
92	89, 90, 91	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
93	92	adantll 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
94		elinel2 4123	. . . . . . . . . . . . 13 ⊢ ((𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
95	93, 94	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
96		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
97	9	ffvelrnda 6828	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
98	97	3adant2 1128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
99		simp2 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
100	99	elioored 42186	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ℝ)
101	98	rexrd 10680	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ^*)
102	15	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐸 ∈ ℝ)
103	76, 20	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ)
104	103	nnred 11640	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ)
105	104	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ)
106		0red 10633	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ∈ ℝ)
107	103	nngt0d 11674	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 < (♯‘𝐼))
108	106, 104, 107	ltled 10777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 ≤ (♯‘𝐼))
109	108	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ≤ (♯‘𝐼))
110	105, 109	resqrtcld 14769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ)
111		sqrtgt0 14610	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐼) ∈ ℝ ∧ 0 < (♯‘𝐼)) → 0 < (√‘(♯‘𝐼)))
112	104, 107, 111	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 < (√‘(♯‘𝐼)))
113	106, 112	gtned 10764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (√‘(♯‘𝐼)) ≠ 0)
114	113	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(♯‘𝐼)) ≠ 0)
115	102, 110, 114	redivcld 11457	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
116	97, 115	readdcld 10659	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
117	116	rexrd 10680	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
118	117	3adant2 1128	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
119		ioogtlb 42132	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋‘𝑖) ∈ ℝ^* ∧ ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^* ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑋‘𝑖) < (𝑦‘𝑖))
120	101, 118, 99, 119	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) < (𝑦‘𝑖))
121	98, 100, 120	ltled 10777	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ≤ (𝑦‘𝑖))
122	98, 100, 121	abssuble0d 14784	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) = ((𝑦‘𝑖) − (𝑋‘𝑖)))
123	116	3adant2 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
124		iooltub 42147	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋‘𝑖) ∈ ℝ^* ∧ ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^* ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑦‘𝑖) < ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
125	101, 118, 99, 124	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) < ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
126	100, 123, 98, 125	ltsub1dd 11241	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑦‘𝑖) − (𝑋‘𝑖)) < (((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) − (𝑋‘𝑖)))
127	98	recnd 10658	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℂ)
128	104, 108	resqrtcld 14769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘(♯‘𝐼)) ∈ ℝ)
129	15, 128, 113	redivcld 11457	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
130	129	recnd 10658	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝐼))) ∈ ℂ)
131	130	3ad2ant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℂ)
132	127, 131	pncan2d 10988	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) − (𝑋‘𝑖)) = (𝐸 / (√‘(♯‘𝐼))))
133	126, 132	breqtrd 5056	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑦‘𝑖) − (𝑋‘𝑖)) < (𝐸 / (√‘(♯‘𝐼))))
134	122, 133	eqbrtrd 5052	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) < (𝐸 / (√‘(♯‘𝐼))))
135	80, 95, 96, 134	syl3anc 1368	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) < (𝐸 / (√‘(♯‘𝐼))))
136	135	adantlrl 719	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) < (𝐸 / (√‘(♯‘𝐼))))
137	14	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐸 ∈ ℝ⁺)
138	104, 107	elrpd 12416	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ⁺)
139	138	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (♯‘𝐼) ∈ ℝ⁺)
140	139	rpsqrtcld 14763	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
141	137, 140	rpdivcld 12436	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ⁺)
142		qndenserrnbllem.d	. . . . . . . . . 10 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
143	75, 77, 78, 79, 74, 136, 141, 142	rrndistlt 42932	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑋𝐷𝑦) < ((√‘(♯‘𝐼)) · (𝐸 / (√‘(♯‘𝐼)))))
144	137	rpcnd 12421	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐸 ∈ ℂ)
145	139	rpcnd 12421	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (♯‘𝐼) ∈ ℂ)
146	145	sqrtcld 14789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ∈ ℂ)
147	140	rpne0d 12424	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ≠ 0)
148	144, 146, 147	divcan2d 11407	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → ((√‘(♯‘𝐼)) · (𝐸 / (√‘(♯‘𝐼)))) = 𝐸)
149	143, 148	breqtrd 5056	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑋𝐷𝑦) < 𝐸)
150	74, 149	jca 515	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑦 ∈ (ℝ ↑_m 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸))
151	142	rrxmetfi 24016	. . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
152	1, 151	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
153		metxmet 22941	. . . . . . . . . 10 ⊢ (𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)) → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)))
154	152, 153	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)))
155	15	rexrd 10680	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
156		elbl 22995	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)) ∧ 𝑋 ∈ (ℝ ↑_m 𝐼) ∧ 𝐸 ∈ ℝ^*) → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ (𝑦 ∈ (ℝ ↑_m 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸)))
157	154, 7, 155, 156	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ (𝑦 ∈ (ℝ ↑_m 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸)))
158	157	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ (𝑦 ∈ (ℝ ↑_m 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸)))
159	150, 158	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
160	68, 159	jca 515	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸)))
161	160	ex 416	. . . 4 ⊢ (𝜑 → ((𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸))))
162	161	eximdv 1918	. . 3 ⊢ (𝜑 → (∃𝑦(𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸))))
163	56, 162	mpd 15	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸)))
164		df-rex 3112	. 2 ⊢ (∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸)))
165	163, 164	sylibr 237	1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑_m cmap 8389 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 ℚcq 12336 ℝ⁺crp 12377 (,)cioo 12726 ♯chash 13686 √csqrt 14584 abscabs 14585 distcds 16566 ∞Metcxmet 20076 Metcmet 20077 ballcbl 20078 ℝ^crrx 23987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-field 19498 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-cnfld 20092 df-refld 20294 df-dsmm 20421 df-frlm 20436 df-nm 23189 df-tng 23191 df-tcph 23774 df-rrx 23989

This theorem is referenced by: qndenserrnbl 42937

W3C validator