qndenserrnbllem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > qndenserrnbllem		Structured version Visualization version GIF version

Theorem qndenserrnbllem 43835

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 4162	. . . . . 6 ⊢ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ
3		qex 12701	. . . . . 6 ⊢ ℚ ∈ V
4		ssexg 5247	. . . . . 6 ⊢ (((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ⊆ ℚ ∧ ℚ ∈ V) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∈ V)
5	2, 3, 4	mp2an 689	. . . . 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 8637	. . . . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
12	10, 11	ffvelrnd 6962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ)
13	12	rexrd 11025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) ∈ ℝ^*)
14		qndenserrnbllem.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
15	14	rpred 12772	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ)
16	15	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ)
17		ne0i 4268	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐼 → 𝐼 ≠ ∅)
18	17	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ≠ ∅)
19		hashnncl 14081	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℕ)
23	22	nnred 11988	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ)
24		0red 10978	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ∈ ℝ)
25	22	nngt0d 12022	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (♯‘𝐼))
26	24, 23, 25	ltled 11123	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 ≤ (♯‘𝐼))
27	23, 26	resqrtcld 15129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ)
28	23, 25	elrpd 12769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ⁺)
29	28	sqrtgt0d 15124	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 0 < (√‘(♯‘𝐼)))
30	24, 29	gtned 11110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ≠ 0)
31	16, 27, 30	redivcld 11803	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
32	12, 31	readdcld 11004	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
33	32	rexrd 11025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
34	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ ℝ⁺)
35	27, 29	elrpd 12769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
36	34, 35	rpdivcld 12789	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ⁺)
37	12, 36	ltaddrpd 12805	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑋‘𝑘) < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))
38		qbtwnxr 12934	. . . . . . . 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 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞 ∈ ℚ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))
40		df-rex 3070	. . . . . . 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 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∃𝑞(𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
42		simprl 768	. . . . . . . . 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 12693	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ℚ → 𝑞 ∈ ℝ)
46	45	ad2antrl 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → 𝑞 ∈ ℝ)
47		simprrl 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑞 ∈ ℚ ∧ ((𝑋‘𝑘) < 𝑞 ∧ 𝑞 < ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) → (𝑋‘𝑘) < 𝑞)
48		simprrr 779	. . . . . . . . . 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 43036	. . . . . . . . 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 4128	. . . . . . . 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 413	. . . . . . 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 1920	. . . . . 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 4280	. . . . 5 ⊢ ((ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ≠ ∅ ↔ ∃𝑞 𝑞 ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))
55	53, 54	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ≠ ∅)
56	1, 6, 55	choicefi 42740	. . 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 3920	. . . . . . . . . . 11 ⊢ (𝑦 Fn 𝐼 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → (𝑦‘𝑘) ∈ ℚ))
59	58	ralimdv 3109	. . . . . . . . . 10 ⊢ (𝑦 Fn 𝐼 → (∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) → ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
60	59	imdistani 569	. . . . . . . . 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 6992	. . . . . . . . 9 ⊢ (𝑦:𝐼⟶ℚ ↔ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ ℚ))
62	60, 61	sylibr 233	. . . . . . . 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 8628	. . . . . . . . 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 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 256	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝑦 ∈ (ℚ ↑_m 𝐼))
69		reex 10962	. . . . . . . . . . 11 ⊢ ℝ ∈ V
70	45	ssriv 3925	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
71		mapss 8677	. . . . . . . . . . 11 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼))
72	69, 70, 71	mp2an 689	. . . . . . . . . 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 3922	. . . . . . . 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 2738	. . . . . . . . . 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 764	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝜑)
81		fveq2 6774	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑦‘𝑘) = (𝑦‘𝑖))
82		fveq2 6774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → (𝑋‘𝑘) = (𝑋‘𝑖))
83	82	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))) = ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
84	82, 83	oveq12d 7293	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))) = ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
85	84	ineq2d 4146	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) = (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))))
86	81, 85	eleq12d 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → ((𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ↔ (𝑦‘𝑖) ∈ (ℚ ∩ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))))
87	86	cbvralvw 3383	. . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
91		rspa 3132	. . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . 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 4130	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼))))))) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
97	9	ffvelrnda 6961	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
98	97	3adant2 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℝ)
99		simp2 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))))
100	99	elioored 43087	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) ∈ ℝ)
101	98	rexrd 11025	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ)
104	103	nnred 11988	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (♯‘𝐼) ∈ ℝ)
105	104	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (♯‘𝐼) ∈ ℝ)
106		0red 10978	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ∈ ℝ)
107	103	nngt0d 12022	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 < (♯‘𝐼))
108	106, 104, 107	ltled 11123	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 ≤ (♯‘𝐼))
109	108	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 0 ≤ (♯‘𝐼))
110	105, 109	resqrtcld 15129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(♯‘𝐼)) ∈ ℝ)
111		sqrtgt0 14970	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐼) ∈ ℝ ∧ 0 < (♯‘𝐼)) → 0 < (√‘(♯‘𝐼)))
112	104, 107, 111	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 < (√‘(♯‘𝐼)))
113	106, 112	gtned 11110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (√‘(♯‘𝐼)) ≠ 0)
114	113	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (√‘(♯‘𝐼)) ≠ 0)
115	102, 110, 114	redivcld 11803	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
116	97, 115	readdcld 11004	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
117	116	rexrd 11025	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
118	117	3adant2 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ^*)
119		ioogtlb 43033	. . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) < (𝑦‘𝑖))
121	98, 100, 120	ltled 11123	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ≤ (𝑦‘𝑖))
122	98, 100, 121	abssuble0d 15144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑦‘𝑖))) = ((𝑦‘𝑖) − (𝑋‘𝑖)))
123	116	3adant2 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) ∈ ℝ)
124		iooltub 43048	. . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑦‘𝑖) < ((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))))
126	100, 123, 98, 125	ltsub1dd 11587	. . . . . . . . . . . . . 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 11003	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝑋‘𝑖) ∈ ℂ)
128	104, 108	resqrtcld 15129	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘(♯‘𝐼)) ∈ ℝ)
129	15, 128, 113	redivcld 11803	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝐼))) ∈ ℝ)
130	129	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝐼))) ∈ ℂ)
131	130	3ad2ant1 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (𝐸 / (√‘(♯‘𝐼))) ∈ ℂ)
132	127, 131	pncan2d 11334	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → (((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼)))) − (𝑋‘𝑖)) = (𝐸 / (√‘(♯‘𝐼))))
133	126, 132	breqtrd 5100	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦‘𝑖) ∈ ((𝑋‘𝑖)(,)((𝑋‘𝑖) + (𝐸 / (√‘(♯‘𝐼))))) ∧ 𝑖 ∈ 𝐼) → ((𝑦‘𝑖) − (𝑋‘𝑖)) < (𝐸 / (√‘(♯‘𝐼))))
134	122, 133	eqbrtrd 5096	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 717	. . . . . . . . . 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 12769	. . . . . . . . . . . . 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 15123	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
141	137, 140	rpdivcld 12789	. . . . . . . . . 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 43831	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑋𝐷𝑦) < ((√‘(♯‘𝐼)) · (𝐸 / (√‘(♯‘𝐼)))))
144	137	rpcnd 12774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → 𝐸 ∈ ℂ)
145	139	rpcnd 12774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (♯‘𝐼) ∈ ℂ)
146	145	sqrtcld 15149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ∈ ℂ)
147	140	rpne0d 12777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (√‘(♯‘𝐼)) ≠ 0)
148	144, 146, 147	divcan2d 11753	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → ((√‘(♯‘𝐼)) · (𝐸 / (√‘(♯‘𝐼)))) = 𝐸)
149	143, 148	breqtrd 5100	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑦‘𝑘) ∈ (ℚ ∩ ((𝑋‘𝑘)(,)((𝑋‘𝑘) + (𝐸 / (√‘(♯‘𝐼)))))))) → (𝑋𝐷𝑦) < 𝐸)
150	74, 149	jca 512	. . . . . . 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 24576	. . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
152	1, 151	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
153		metxmet 23487	. . . . . . . . . 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 11025	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
156		elbl 23541	. . . . . . . . 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 1370	. . . . . . . 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 256	. . . . . 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 512	. . . . 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 413	. . . 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 1920	. . 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 3070	. 2 ⊢ (∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ (𝑋(ball‘𝐷)𝐸)))
165	163, 164	sylibr 233	1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 + caddc 10874 · cmul 10876 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 ℚcq 12688 ℝ⁺crp 12730 (,)cioo 13079 ♯chash 14044 √csqrt 14944 abscabs 14945 distcds 16971 ∞Metcxmet 20582 Metcmet 20583 ballcbl 20584 ℝ^crrx 24547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xadd 12849 df-ioo 13083 df-ico 13085 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-field 19994 df-subrg 20022 df-staf 20105 df-srng 20106 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-cnfld 20598 df-refld 20810 df-dsmm 20939 df-frlm 20954 df-nm 23738 df-tng 23740 df-tcph 24333 df-rrx 24549

This theorem is referenced by: qndenserrnbl 43836

W3C validator