qndenserrnbllem - Mathbox for Glauco Siliprandi

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Theorem qndenserrnbllem 46309

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 4237	. . . . . 6 ⊢ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ
3		qex 13003	. . . . . 6 ⊢ ℚ ∈ V
4		ssexg 5323	. . . . . 6 ⊢ (((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ ∧ ℚ ∈ V) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∈ V)
5	2, 3, 4	mp2an 692	. . . . 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 8889	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → 𝑋:𝐼⟶ℝ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋:𝐼⟶ℝ)
10	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑋:𝐼⟶ℝ)
11		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
12	10, 11	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ)
13	12	rexrd 11311	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ^*)
14		qndenserrnbllem.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
15	14	rpred 13077	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ)
16	15	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ)
17		ne0i 4341	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐼 → 𝐼 ≠ ∅)
18	17	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ≠ ∅)
19		hashnncl 14405	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ Fin → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
20	1, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
21	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
22	18, 21	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℕ)
23	22	nnred 12281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ)
24		0red 11264	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ∈ ℝ)
25	22	nngt0d 12315	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (♯‘𝐼))
26	24, 23, 25	ltled 11409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ≤ (♯‘𝐼))
27	23, 26	resqrtcld 15456	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ)
28	23, 25	elrpd 13074	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ⁺)
29	28	sqrtgt0d 15451	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (√‘(♯‘𝐼)))
30	24, 29	gtned 11396	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ≠ 0)
31	16, 27, 30	redivcld 12095	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
32	12, 31	readdcld 11290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
33	32	rexrd 11311	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
34	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ⁺)
35	27, 29	elrpd 13074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
36	34, 35	rpdivcld 13094	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ⁺)
37	12, 36	ltaddrpd 13110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))
38		qbtwnxr 13242	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞 ∈ ℚ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))
40		df-rex 3071	. . . . . . 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 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞(𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
42		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ ℚ)
43	13	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑋‘𝑘) ∈ ℝ^*)
44	33	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
45		qre 12995	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ℚ → 𝑞 ∈ ℝ)
46	45	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ ℝ)
47		simprrl 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑋‘𝑘) < 𝑞)
48		simprrr 782	. . . . . . . . . 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 45511	. . . . . . . . 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 4200	. . . . . . . 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 412	. . . . . . 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 1917	. . . . . 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 4353	. . . . 5 ⊢ ((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ≠ ∅ ↔ ∃𝑞 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
55	53, 54	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ≠ ∅)
56	1, 6, 55	choicefi 45205	. . 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 3982	. . . . . . . . . . 11 ⊢ (𝑦 Fn 𝐼 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑦‘𝑘) ∈ ℚ))
59	58	ralimdv 3169	. . . . . . . . . 10 ⊢ (𝑦 Fn 𝐼 → (∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
60	59	imdistani 568	. . . . . . . . 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 7139	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℚ ↔ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
62	60, 61	sylibr 234	. . . . . . . 8 ⊢ ((𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑦:𝐼⟶ℚ)
63	62	adantl 481	. . . . . . 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 8879	. . . . . . . . 9 ⊢ ((ℚ ∈ V ∧ 𝐼 ∈ Fin) → (𝑦 ∈ (ℚ ↑_m 𝐼) ↔ 𝑦:𝐼⟶ℚ))
66	64, 1, 65	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (ℚ ↑_m 𝐼) ↔ 𝑦:𝐼⟶ℚ))
67	66	adantr 480	. . . . . . 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 257	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦 ∈ (ℚ ↑_m 𝐼))
69		reex 11246	. . . . . . . . . . 11 ⊢ ℝ ∈ V
70	45	ssriv 3987	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
71		mapss 8929	. . . . . . . . . . 11 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼))
72	69, 70, 71	mp2an 692	. . . . . . . . . 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 3984	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦 ∈ (ℝ ↑_m 𝐼))
75	1	adantr 480	. . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐼 ≠ ∅)
78		eqid 2737	. . . . . . . . . 10 ⊢ (♯‘𝐼) = (♯‘𝐼)
79	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑋 ∈ (ℝ ↑_m 𝐼))
80		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝜑)
81		fveq2 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑦‘𝑘) = (𝑦‘𝑖))
82		fveq2 6906	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → (𝑋‘𝑘) = (𝑋‘𝑖))
83	82	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) = ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
84	82, 83	oveq12d 7449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))) = ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
85	84	ineq2d 4220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) = (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
86	81, 85	eleq12d 2835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ↔ (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))))
87	86	cbvralvw 3237	. . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
91		rspa 3248	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 714	. . . . . . . . . . . . 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 4202	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
97	9	ffvelcdmda 7104	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
98	97	3adant2 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
99		simp2 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
100	99	elioored 45562	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ℝ)
101	98	rexrd 11311	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ^*)
102	15	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐸 ∈ ℝ)
103	76, 20	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ)
104	103	nnred 12281	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ)
105	104	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ)
106		0red 11264	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ∈ ℝ)
107	103	nngt0d 12315	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 < (♯‘𝐼))
108	106, 104, 107	ltled 11409	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 ≤ (♯‘𝐼))
109	108	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ≤ (♯‘𝐼))
110	105, 109	resqrtcld 15456	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ)
111		sqrtgt0 15297	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐼) ∈ ℝ ∧ 0 < (♯‘𝐼)) → 0 < (√‘(♯‘𝐼)))
112	104, 107, 111	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 < (√‘(♯‘𝐼)))
113	106, 112	gtned 11396	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (√‘(♯‘𝐼)) ≠ 0)
114	113	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(♯‘𝐼)) ≠ 0)
115	102, 110, 114	redivcld 12095	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
116	97, 115	readdcld 11290	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
117	116	rexrd 11311	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
118	117	3adant2 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
119		ioogtlb 45508	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) < (𝑦‘𝑖))
121	98, 100, 120	ltled 11409	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ≤ (𝑦‘𝑖))
122	98, 100, 121	abssuble0d 15471	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) = ((𝑦‘𝑖) − (𝑋‘𝑖)))
123	116	3adant2 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
124		iooltub 45523	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) < ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
126	100, 123, 98, 125	ltsub1dd 11875	. . . . . . . . . . . . . 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 11289	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℂ)
128	104, 108	resqrtcld 15456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘(♯‘𝐼)) ∈ ℝ)
129	15, 128, 113	redivcld 12095	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
130	129	recnd 11289	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝐼))) ∈ ℂ)
131	130	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℂ)
132	127, 131	pncan2d 11622	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) − (𝑋‘𝑖)) = (𝐸 / (√‘(♯‘𝐼))))
133	126, 132	breqtrd 5169	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑦‘𝑖) − (𝑋‘𝑖)) < (𝐸 / (√‘(♯‘𝐼))))
134	122, 133	eqbrtrd 5165	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 720	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐸 ∈ ℝ⁺)
138	104, 107	elrpd 13074	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ⁺)
139	138	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (♯‘𝐼) ∈ ℝ⁺)
140	139	rpsqrtcld 15450	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
141	137, 140	rpdivcld 13094	. . . . . . . . . 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 46305	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑋𝐷𝑦) < ((√‘(♯‘𝐼)) · (𝐸 / (√‘(♯‘𝐼)))))
144	137	rpcnd 13079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐸 ∈ ℂ)
145	139	rpcnd 13079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (♯‘𝐼) ∈ ℂ)
146	145	sqrtcld 15476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ∈ ℂ)
147	140	rpne0d 13082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ≠ 0)
148	144, 146, 147	divcan2d 12045	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → ((√‘(♯‘𝐼)) · (𝐸 / (√‘(♯‘𝐼)))) = 𝐸)
149	143, 148	breqtrd 5169	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑋𝐷𝑦) < 𝐸)
150	74, 149	jca 511	. . . . . . 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 25446	. . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
152	1, 151	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
153		metxmet 24344	. . . . . . . . . 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 11311	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
156		elbl 24398	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ (𝑦 ∈ (ℝ ↑_m 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸)))
158	157	adantr 480	. . . . . . 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 257	. . . . . 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 511	. . . . 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 412	. . . 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 1917	. . 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 3071	. 2 ⊢ (∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸)))
165	163, 164	sylibr 234	1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑_m cmap 8866 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 + caddc 11158 · cmul 11160 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 ℚcq 12990 ℝ⁺crp 13034 (,)cioo 13387 ♯chash 14369 √csqrt 15272 abscabs 15273 distcds 17306 ∞Metcxmet 21349 Metcmet 21350 ballcbl 21351 ℝ^crrx 25417

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-field 20732 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-cnfld 21365 df-refld 21623 df-dsmm 21752 df-frlm 21767 df-nm 24595 df-tng 24597 df-tcph 25203 df-rrx 25419

This theorem is referenced by: qndenserrnbl 46310

W3C validator