qndenserrnbllem - Mathbox for Glauco Siliprandi

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Theorem qndenserrnbllem 47577

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 4136	. . . . . 6 ⊢ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ
3		qex 13128	. . . . . 6 ⊢ ℚ ∈ V
4		ssexg 5288	. . . . . 6 ⊢ (((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ ∧ ℚ ∈ V) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∈ V)
5	2, 3, 4	mp2an 705	. . . . 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 8907	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → 𝑋:𝐼⟶ℝ)
9	7, 8	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋:𝐼⟶ℝ)
10	9	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑋:𝐼⟶ℝ)
11		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
12	10, 11	ffvelcdmd 7117	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ)
13	12	rexrd 11398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ^*)
14		qndenserrnbllem.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
15	14	rpred 13204	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ)
16	15	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ)
17		ne0i 4290	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐼 → 𝐼 ≠ ∅)
18	17	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ≠ ∅)
19		hashnncl 14551	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ Fin → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
20	1, 19	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
21	20	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
22	18, 21	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℕ)
23	22	nnred 12390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ)
24		0red 11350	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ∈ ℝ)
25	22	nngt0d 12427	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (♯‘𝐼))
26	24, 23, 25	ltled 11497	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ≤ (♯‘𝐼))
27	23, 26	resqrtcld 15629	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ)
28	23, 25	elrpd 13201	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ⁺)
29	28	sqrtgt0d 15624	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (√‘(♯‘𝐼)))
30	24, 29	gtned 11484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ≠ 0)
31	16, 27, 30	redivcld 12185	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
32	12, 31	readdcld 11377	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
33	32	rexrd 11398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
34	14	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ⁺)
35	27, 29	elrpd 13201	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
36	34, 35	rpdivcld 13221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ⁺)
37	12, 36	ltaddrpd 13237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))
38		qbtwnxr 13370	. . . . . . . 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 1399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞 ∈ ℚ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))
40		df-rex 3091	. . . . . . 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 783	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ ℚ)
43	13	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑋‘𝑘) ∈ ℝ^*)
44	33	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
45		qre 13120	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ℚ → 𝑞 ∈ ℝ)
46	45	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ ℝ)
47		simprrl 793	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑋‘𝑘) < 𝑞)
48		simprrr 794	. . . . . . . . . 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 46783	. . . . . . . . 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 4100	. . . . . . . 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 418	. . . . . . 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 1952	. . . . . 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 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
54		n0 4303	. . . . 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 46487	. . 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 3912	. . . . . . . . . . 11 ⊢ (𝑦 Fn 𝐼 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑦‘𝑘) ∈ ℚ))
59	58	ralimdv 3180	. . . . . . . . . 10 ⊢ (𝑦 Fn 𝐼 → (∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
60	59	imdistani 579	. . . . . . . . 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 7151	. . . . . . . . 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 487	. . . . . . 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 8897	. . . . . . . . 9 ⊢ ((ℚ ∈ V ∧ 𝐼 ∈ Fin) → (𝑦 ∈ (ℚ ↑_m 𝐼) ↔ 𝑦:𝐼⟶ℚ))
66	64, 1, 65	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (ℚ ↑_m 𝐼) ↔ 𝑦:𝐼⟶ℚ))
67	66	adantr 486	. . . . . . 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 11330	. . . . . . . . . . 11 ⊢ ℝ ∈ V
70	45	ssriv 3917	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
71		mapss 8955	. . . . . . . . . . 11 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼))
72	69, 70, 71	mp2an 705	. . . . . . . . . 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 3914	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦 ∈ (ℝ ↑_m 𝐼))
75	1	adantr 486	. . . . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐼 ≠ ∅)
78		eqid 2763	. . . . . . . . . 10 ⊢ (♯‘𝐼) = (♯‘𝐼)
79	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑋 ∈ (ℝ ↑_m 𝐼))
80		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝜑)
81		fveq2 6916	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝑦‘𝑘) = (𝑦‘𝑖))
82		fveq2 6916	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → (𝑋‘𝑘) = (𝑋‘𝑖))
83	82	oveq1d 7466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) = ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
84	82, 83	oveq12d 7469	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))) = ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
85	84	ineq2d 4120	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) = (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
86	81, 85	eleq12d 2858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ↔ (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))))
87	86	cbvralvw 3244	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ↔ ∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
88	87	birani 509	. . . . . . . . . . . . . . 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)))))))
89		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
90		rspa 3255	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑖 ∈ 𝐼 (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
91	88, 89, 90	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
92	91	adantll 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
93		elinel2 4102	. . . . . . . . . . . . 13 ⊢ ((𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
94	92, 93	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
95		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
96	9	ffvelcdmda 7116	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
97	96	3adant2 1149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
98		simp2 1155	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
99	98	elioored 46834	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ℝ)
100	97	rexrd 11398	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ^*)
101	15	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐸 ∈ ℝ)
102	76, 20	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ)
103	102	nnred 12390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ)
104	103	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ)
105		0red 11350	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ∈ ℝ)
106	102	nngt0d 12427	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 < (♯‘𝐼))
107	105, 103, 106	ltled 11497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 ≤ (♯‘𝐼))
108	107	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ≤ (♯‘𝐼))
109	104, 108	resqrtcld 15629	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ)
110		sqrtgt0 15468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐼) ∈ ℝ ∧ 0 < (♯‘𝐼)) → 0 < (√‘(♯‘𝐼)))
111	103, 106, 110	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 < (√‘(♯‘𝐼)))
112	105, 111	gtned 11484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (√‘(♯‘𝐼)) ≠ 0)
113	112	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(♯‘𝐼)) ≠ 0)
114	101, 109, 113	redivcld 12185	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
115	96, 114	readdcld 11377	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
116	115	rexrd 11398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
117	116	3adant2 1149	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
118		ioogtlb 46780	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋‘𝑖) ∈ ℝ^* ∧ ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^* ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑋‘𝑖) < (𝑦‘𝑖))
119	100, 117, 98, 118	syl3anc 1399	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) < (𝑦‘𝑖))
120	97, 99, 119	ltled 11497	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ≤ (𝑦‘𝑖))
121	97, 99, 120	abssuble0d 15646	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) = ((𝑦‘𝑖) − (𝑋‘𝑖)))
122	115	3adant2 1149	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
123		iooltub 46795	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋‘𝑖) ∈ ℝ^* ∧ ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^* ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑦‘𝑖) < ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
124	100, 117, 98, 123	syl3anc 1399	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) < ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
125	99, 122, 97, 124	ltsub1dd 11968	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑦‘𝑖) − (𝑋‘𝑖)) < (((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) − (𝑋‘𝑖)))
126	97	recnd 11376	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℂ)
127	103, 107	resqrtcld 15629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘(♯‘𝐼)) ∈ ℝ)
128	15, 127, 112	redivcld 12185	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
129	128	recnd 11376	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝐼))) ∈ ℂ)
130	129	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℂ)
131	126, 130	pncan2d 11710	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) − (𝑋‘𝑖)) = (𝐸 / (√‘(♯‘𝐼))))
132	125, 131	breqtrd 5137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑦‘𝑖) − (𝑋‘𝑖)) < (𝐸 / (√‘(♯‘𝐼))))
133	121, 132	eqbrtrd 5133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) < (𝐸 / (√‘(♯‘𝐼))))
134	80, 94, 95, 133	syl3anc 1399	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) < (𝐸 / (√‘(♯‘𝐼))))
135	134	adantlrl 733	. . . . . . . . . 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))))
136	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐸 ∈ ℝ⁺)
137	103, 106	elrpd 13201	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ⁺)
138	137	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (♯‘𝐼) ∈ ℝ⁺)
139	138	rpsqrtcld 15623	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
140	136, 139	rpdivcld 13221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ⁺)
141		qndenserrnbllem.d	. . . . . . . . . 10 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
142	75, 77, 78, 79, 74, 135, 140, 141	rrndistlt 47573	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑋𝐷𝑦) < ((√‘(♯‘𝐼)) · (𝐸 / (√‘(♯‘𝐼)))))
143	136	rpcnd 13206	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐸 ∈ ℂ)
144	138	rpcnd 13206	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (♯‘𝐼) ∈ ℂ)
145	144	sqrtcld 15651	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ∈ ℂ)
146	139	rpne0d 13209	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ≠ 0)
147	143, 145, 146	divcan2d 12135	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → ((√‘(♯‘𝐼)) · (𝐸 / (√‘(♯‘𝐼)))) = 𝐸)
148	142, 147	breqtrd 5137	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑋𝐷𝑦) < 𝐸)
149	74, 148	jca 521	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑦 ∈ (ℝ ↑_m 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸))
150	141	rrxmetfi 25788	. . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
151	1, 150	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
152		metxmet 24708	. . . . . . . . . 10 ⊢ (𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)) → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)))
153	151, 152	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)))
154	15	rexrd 11398	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
155		elbl 24762	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)) ∧ 𝑋 ∈ (ℝ ↑_m 𝐼) ∧ 𝐸 ∈ ℝ^*) → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ (𝑦 ∈ (ℝ ↑_m 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸)))
156	153, 7, 154, 155	syl3anc 1399	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ (𝑦 ∈ (ℝ ↑_m 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸)))
157	156	adantr 486	. . . . . . 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)))
158	149, 157	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
159	68, 158	jca 521	. . . . 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)))
160	159	ex 418	. . . 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))))
161	160	eximdv 1952	. . 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))))
162	56, 161	mpd 16	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸)))
163		df-rex 3091	. 2 ⊢ (∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸)))
164	162, 163	sylibr 237	1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 Vcvv 3454 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 class class class wbr 5108 Fn wfn 6561 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 Fincfn 9011 ℂcc 11237 ℝcr 11238 0cc0 11239 + caddc 11242 · cmul 11244 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 − cmin 11580 / cdiv 12013 ℕcn 12375 ℚcq 13115 ℝ⁺crp 13160 (,)cioo 13516 ♯chash 14515 √csqrt 15443 abscabs 15444 distcds 17481 ∞Metcxmet 21714 Metcmet 21715 ballcbl 21716 ℝ^crrx 25759

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318 ax-mulf 11319

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-tpos 8281 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xadd 13282 df-ioo 13520 df-ico 13522 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-sum 15898 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-0g 17662 df-gsum 17663 df-prds 17668 df-pws 17670 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-grp 19197 df-minusg 19198 df-sbg 19199 df-subg 19383 df-ghm 19478 df-cntz 19582 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-cring 20513 df-oppr 20618 df-dvdsr 20638 df-unit 20639 df-invr 20669 df-dvr 20682 df-rhm 20753 df-subrng 20849 df-subrg 20873 df-drng 21033 df-field 21034 df-staf 21147 df-srng 21148 df-lmod 21188 df-lss 21258 df-sra 21499 df-rgmod 21500 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-cnfld 21730 df-refld 21966 df-dsmm 22093 df-frlm 22108 df-nm 24956 df-tng 24958 df-tcph 25545 df-rrx 25761

This theorem is used by: qndenserrnbl 47578

W3C validator