qndenserrnbllem - Mathbox for Glauco Siliprandi

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Theorem qndenserrnbllem 40514

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 𝑘 𝑖 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qndenserrnbllem.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
2		inss1 3833	. . . . . 6 ⊢ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) ⊆ ℚ
3		qex 11800	. . . . . 6 ⊢ ℚ ∈ V
4		ssexg 4804	. . . . . 6 ⊢ (((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) ⊆ ℚ ∧ ℚ ∈ V) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) ∈ V)
5	2, 3, 4	mp2an 708	. . . . 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 7879	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (ℝ ↑_𝑚 𝐼) → 𝑋:𝐼⟶ℝ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋:𝐼⟶ℝ)
10	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑋:𝐼⟶ℝ)
11		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
12	10, 11	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ)
13	12	rexrd 10089	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ^*)
14		qndenserrnbllem.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
15	14	rpred 11872	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ)
16	15	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ)
17		ne0i 3921	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐼 → 𝐼 ≠ ∅)
18	17	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ≠ ∅)
19		hashnncl 13157	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ Fin → ((#‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
20	1, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
21	20	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((#‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
22	18, 21	mpbird 247	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (#‘𝐼) ∈ ℕ)
23	22	nnred 11035	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (#‘𝐼) ∈ ℝ)
24		0red 10041	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ∈ ℝ)
25	22	nngt0d 11064	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (#‘𝐼))
26	24, 23, 25	ltled 10185	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ≤ (#‘𝐼))
27	23, 26	resqrtcld 14156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(#‘𝐼)) ∈ ℝ)
28	23, 25	elrpd 11869	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (#‘𝐼) ∈ ℝ⁺)
29	28	sqrtgt0d 14151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (√‘(#‘𝐼)))
30	24, 29	gtned 10172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(#‘𝐼)) ≠ 0)
31	16, 27, 30	redivcld 10853	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(#‘𝐼))) ∈ ℝ)
32	12, 31	readdcld 10069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))) ∈ ℝ)
33	32	rexrd 10089	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))) ∈ ℝ^*)
34	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ⁺)
35	27, 29	elrpd 11869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(#‘𝐼)) ∈ ℝ⁺)
36	34, 35	rpdivcld 11889	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(#‘𝐼))) ∈ ℝ⁺)
37	12, 36	ltaddrpd 11905	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) < ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))
38		qbtwnxr 12031	. . . . . . . 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 1326	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞 ∈ ℚ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))
40		df-rex 2918	. . . . . . 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 208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞(𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))
42		simprl 794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))) → 𝑞 ∈ ℚ)
43	13	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))) → (𝑋‘𝑘) ∈ ℝ^*)
44	33	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))) → ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))) ∈ ℝ^*)
45		qre 11793	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ℚ → 𝑞 ∈ ℝ)
46	45	ad2antrl 764	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))) → 𝑞 ∈ ℝ)
47		simprrl 804	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))) → (𝑋‘𝑘) < 𝑞)
48		simprrr 805	. . . . . . . . . 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 39720	. . . . . . . . 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 3798	. . . . . . . 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 450	. . . . . . 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 1846	. . . . . 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 3931	. . . . 5 ⊢ ((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) ≠ ∅ ↔ ∃𝑞 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))
55	53, 54	sylibr 224	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) ≠ ∅)
56	1, 6, 55	choicefi 39392	. . 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 3602	. . . . . . . . . . 11 ⊢ (𝑦 Fn 𝐼 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) → (𝑦‘𝑘) ∈ ℚ))
59	58	ralimdv 2963	. . . . . . . . . 10 ⊢ (𝑦 Fn 𝐼 → (∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) → ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
60	59	imdistani 726	. . . . . . . . 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 6388	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℚ ↔ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
62	60, 61	sylibr 224	. . . . . . . 8 ⊢ ((𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))) → 𝑦:𝐼⟶ℚ)
63	62	adantl 482	. . . . . . 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 7870	. . . . . . . . 9 ⊢ ((ℚ ∈ V ∧ 𝐼 ∈ Fin) → (𝑦 ∈ (ℚ ↑_𝑚 𝐼) ↔ 𝑦:𝐼⟶ℚ))
66	64, 1, 65	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (ℚ ↑_𝑚 𝐼) ↔ 𝑦:𝐼⟶ℚ))
67	66	adantr 481	. . . . . . 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 247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → 𝑦 ∈ (ℚ ↑_𝑚 𝐼))
69		reex 10027	. . . . . . . . . . 11 ⊢ ℝ ∈ V
70	45	ssriv 3607	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
71		mapss 7900	. . . . . . . . . . 11 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑_𝑚 𝐼) ⊆ (ℝ ↑_𝑚 𝐼))
72	69, 70, 71	mp2an 708	. . . . . . . . . 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 3604	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → 𝑦 ∈ (ℝ ↑_𝑚 𝐼))
75	1	adantr 481	. . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → 𝐼 ≠ ∅)
78		eqid 2622	. . . . . . . . . 10 ⊢ (#‘𝐼) = (#‘𝐼)
79	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → 𝑋 ∈ (ℝ ↑_𝑚 𝐼))
80		simpll 790	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝜑)
81		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑦‘𝑘) = (𝑦‘𝑖))
82		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → (𝑋‘𝑘) = (𝑋‘𝑖))
83	82	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))) = ((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼)))))
84	82, 83	oveq12d 6668	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))) = ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))))
85	84	ineq2d 3814	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) = (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼)))))))
86	81, 85	eleq12d 2695	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) ↔ (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))))))
87	86	cbvralv 3171	. . . . . . . . . . . . . . . . 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 206	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
91		rspa 2930	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 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 750	. . . . . . . . . . . . 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 3800	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
97	9	ffvelrnda 6359	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
98	97	3adant2 1080	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
99		simp2 1062	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))))
100	99	elioored 39776	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ℝ)
101	98	rexrd 10089	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ^*)
102	15	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐸 ∈ ℝ)
103	76, 20	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (#‘𝐼) ∈ ℕ)
104	103	nnred 11035	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (#‘𝐼) ∈ ℝ)
105	104	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (#‘𝐼) ∈ ℝ)
106		0red 10041	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ∈ ℝ)
107	103	nngt0d 11064	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 < (#‘𝐼))
108	106, 104, 107	ltled 10185	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 ≤ (#‘𝐼))
109	108	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ≤ (#‘𝐼))
110	105, 109	resqrtcld 14156	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(#‘𝐼)) ∈ ℝ)
111		sqrtgt0 13999	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((#‘𝐼) ∈ ℝ ∧ 0 < (#‘𝐼)) → 0 < (√‘(#‘𝐼)))
112	104, 107, 111	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 < (√‘(#‘𝐼)))
113	106, 112	gtned 10172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (√‘(#‘𝐼)) ≠ 0)
114	113	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(#‘𝐼)) ≠ 0)
115	102, 110, 114	redivcld 10853	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(#‘𝐼))) ∈ ℝ)
116	97, 115	readdcld 10069	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼)))) ∈ ℝ)
117	116	rexrd 10089	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼)))) ∈ ℝ^*)
118	117	3adant2 1080	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼)))) ∈ ℝ^*)
119		ioogtlb 39717	. . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) < (𝑦‘𝑖))
121	98, 100, 120	ltled 10185	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ≤ (𝑦‘𝑖))
122	98, 100, 121	abssuble0d 14171	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) = ((𝑦‘𝑖) − (𝑋‘𝑖)))
123	116	3adant2 1080	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼)))) ∈ ℝ)
124		iooltub 39735	. . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) < ((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼)))))
126	100, 123, 98, 125	ltsub1dd 10639	. . . . . . . . . . . . . 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 10068	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℂ)
128	104, 108	resqrtcld 14156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘(#‘𝐼)) ∈ ℝ)
129	15, 128, 113	redivcld 10853	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 / (√‘(#‘𝐼))) ∈ ℝ)
130	129	recnd 10068	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 / (√‘(#‘𝐼))) ∈ ℂ)
131	130	3ad2ant1 1082	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(#‘𝐼))) ∈ ℂ)
132	127, 131	pncan2d 10394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼)))) − (𝑋‘𝑖)) = (𝐸 / (√‘(#‘𝐼))))
133	126, 132	breqtrd 4679	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(#‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑦‘𝑖) − (𝑋‘𝑖)) < (𝐸 / (√‘(#‘𝐼))))
134	122, 133	eqbrtrd 4675	. . . . . . . . . . . 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 1326	. . . . . . . . . . 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 756	. . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → 𝐸 ∈ ℝ⁺)
138	104, 107	elrpd 11869	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘𝐼) ∈ ℝ⁺)
139	138	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → (#‘𝐼) ∈ ℝ⁺)
140	139	rpsqrtcld 14150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → (√‘(#‘𝐼)) ∈ ℝ⁺)
141	137, 140	rpdivcld 11889	. . . . . . . . . 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 40510	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → (𝑋𝐷𝑦) < ((√‘(#‘𝐼)) · (𝐸 / (√‘(#‘𝐼)))))
144	137	rpcnd 11874	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → 𝐸 ∈ ℂ)
145	139	rpcnd 11874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → (#‘𝐼) ∈ ℂ)
146	145	sqrtcld 14176	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → (√‘(#‘𝐼)) ∈ ℂ)
147	140	rpne0d 11877	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → (√‘(#‘𝐼)) ≠ 0)
148	144, 146, 147	divcan2d 10803	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → ((√‘(#‘𝐼)) · (𝐸 / (√‘(#‘𝐼)))) = 𝐸)
149	143, 148	breqtrd 4679	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(#‘𝐼)))))))) → (𝑋𝐷𝑦) < 𝐸)
150	74, 149	jca 554	. . . . . . 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 40507	. . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_𝑚 𝐼)))
152	1, 151	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_𝑚 𝐼)))
153		metxmet 22139	. . . . . . . . . 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 10089	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
156		elbl 22193	. . . . . . . . 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 1326	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ (𝑦 ∈ (ℝ ↑_𝑚 𝐼) ∧ (𝑋𝐷𝑦) < 𝐸)))
158	157	adantr 481	. . . . . . 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 247	. . . . . 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 554	. . . . 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 450	. . . 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 1846	. . 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 2918	. 2 ⊢ (∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∃𝑦(𝑦 ∈ (ℚ ↑_𝑚 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸)))
165	163, 164	sylibr 224	1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 · cmul 9941 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℚcq 11788 ℝ⁺crp 11832 (,)cioo 12175 #chash 13117 √csqrt 13973 abscabs 13974 distcds 15950 ∞Metcxmt 19731 Metcme 19732 ballcbl 19733 ℝ^crrx 23171

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-field 18750 df-subrg 18778 df-staf 18845 df-srng 18846 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-cnfld 19747 df-refld 19951 df-dsmm 20076 df-frlm 20091 df-nm 22387 df-tng 22389 df-tch 22969 df-rrx 23173

This theorem is referenced by: qndenserrnbl 40515

W3C validator