hoiqssbllem3 - Mathbox for Glauco Siliprandi

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Theorem hoiqssbllem3 47455

Description: A n-dimensional ball contains a nonempty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
hoiqssbllem3.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoiqssbllem3.n	⊢ (𝜑 → 𝑋 ≠ ∅)
hoiqssbllem3.y	⊢ (𝜑 → 𝑌 ∈ (ℝ ↑_m 𝑋))
hoiqssbllem3.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)

**Assertion**
Ref	Expression
hoiqssbllem3	⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))

Distinct variable groups: 𝐸,𝑐,𝑑,𝑖 𝑋,𝑐,𝑑,𝑖 𝑌,𝑐,𝑑,𝑖 𝜑,𝑐,𝑑,𝑖

Proof of Theorem hoiqssbllem3 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoiqssbllem3.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		qex 13013	. . . . . . . . 9 ⊢ ℚ ∈ V
3	2	inex1 5280	. . . . . . . 8 ⊢ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∈ V
4	3	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∈ V)
5		hoiqssbllem3.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑_m 𝑋))
6		elmapi 8851	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌:𝑋⟶ℝ)
7	5, 6	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
8	7	ffvelcdmda 7078	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ)
9		hoiqssbllem3.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
10		2rp 13050	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ⁺
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ⁺)
12		hoiqssbllem3.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
13		hashnncl 14433	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
14	1, 13	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
15	12, 14	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
16		nnrp 13057	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑋) ∈ ℕ → (♯‘𝑋) ∈ ℝ⁺)
17	15, 16	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
18	17	rpsqrtcld 15502	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
19	11, 18	rpmulcld 13105	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
20	9, 19	rpdivcld 13106	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
21	20	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
22	8, 21	ltsubrpd 13121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝑌‘𝑖))
23	21	rpred 13089	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
24	8, 23	resubcld 11669	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
25	24, 8	ltnled 11384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝑌‘𝑖) ↔ ¬ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))))
26	22, 25	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))))
27	24	rexrd 11286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
28	8	rexrd 11286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
29	27, 28	qinioo 46368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = ∅ ↔ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))))
30	26, 29	mtbird 328	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = ∅)
31	30	neqned 2962	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ≠ ∅)
32	1, 4, 31	choicefi 46034	. . . . . 6 ⊢ (𝜑 → ∃𝑐(𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))))
33		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))) → 𝑐 Fn 𝑋)
34		nfra1 3286	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
35		rspa 3251	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))
36		elinel1 4147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝑐‘𝑖) ∈ ℚ)
37	35, 36	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ ℚ)
38	37	ex 418	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝑖 ∈ 𝑋 → (𝑐‘𝑖) ∈ ℚ))
39	34, 38	ralrimi 3260	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ)
40	39	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))) → ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ)
41	33, 40	jca 521	. . . . . . . . . . . 12 ⊢ ((𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))) → (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ))
42	41	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ))
43		ffnfv 7113	. . . . . . . . . . 11 ⊢ (𝑐:𝑋⟶ℚ ↔ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ))
44	42, 43	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → 𝑐:𝑋⟶ℚ)
45	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℚ ∈ V)
46		elmapg 8841	. . . . . . . . . . . 12 ⊢ ((ℚ ∈ V ∧ 𝑋 ∈ Fin) → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐:𝑋⟶ℚ))
47	45, 1, 46	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐:𝑋⟶ℚ))
48	47	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐:𝑋⟶ℚ))
49	44, 48	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → 𝑐 ∈ (ℚ ↑_m 𝑋))
50		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))
51	49, 50	jca 521	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))))
52	51	ex 418	. . . . . . 7 ⊢ (𝜑 → ((𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))) → (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))))
53	52	eximdv 1950	. . . . . 6 ⊢ (𝜑 → (∃𝑐(𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))) → ∃𝑐(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))))
54	32, 53	mpd 16	. . . . 5 ⊢ (𝜑 → ∃𝑐(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))))
55		df-rex 3087	. . . . 5 ⊢ (∃𝑐 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ↔ ∃𝑐(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))))
56	54, 55	sylibr 237	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))
57	2	inex1 5280	. . . . . . . 8 ⊢ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ∈ V
58	57	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ∈ V)
59	8, 21	ltaddrpd 13122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
60	8, 23	readdcld 11265	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
61	8, 60	ltnled 11384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ↔ ¬ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖)))
62	59, 61	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖))
63	60	rexrd 11286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
64	28, 63	qinioo 46368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = ∅ ↔ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖)))
65	62, 64	mtbird 328	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = ∅)
66	65	neqned 2962	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ≠ ∅)
67	1, 58, 66	choicefi 46034	. . . . . 6 ⊢ (𝜑 → ∃𝑑(𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
68		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → 𝑑 Fn 𝑋)
69		nfra1 3286	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
70		rspa 3251	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
71		elinel1 4147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝑑‘𝑖) ∈ ℚ)
72	70, 71	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ ℚ)
73	72	ex 418	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝑖 ∈ 𝑋 → (𝑑‘𝑖) ∈ ℚ))
74	69, 73	ralrimi 3260	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ)
75	74	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ)
76	68, 75	jca 521	. . . . . . . . . . . 12 ⊢ ((𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ))
77	76	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ))
78		ffnfv 7113	. . . . . . . . . . 11 ⊢ (𝑑:𝑋⟶ℚ ↔ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ))
79	77, 78	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑:𝑋⟶ℚ)
80		elmapg 8841	. . . . . . . . . . . 12 ⊢ ((ℚ ∈ V ∧ 𝑋 ∈ Fin) → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑:𝑋⟶ℚ))
81	45, 1, 80	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑:𝑋⟶ℚ))
82	81	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑:𝑋⟶ℚ))
83	79, 82	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑 ∈ (ℚ ↑_m 𝑋))
84		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
85	83, 84	jca 521	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → (𝑑 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
86	85	ex 418	. . . . . . 7 ⊢ (𝜑 → ((𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → (𝑑 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))))
87	86	eximdv 1950	. . . . . 6 ⊢ (𝜑 → (∃𝑑(𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → ∃𝑑(𝑑 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))))
88	67, 87	mpd 16	. . . . 5 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
89		df-rex 3087	. . . . 5 ⊢ (∃𝑑 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ↔ ∃𝑑(𝑑 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
90	88, 89	sylibr 237	. . . 4 ⊢ (𝜑 → ∃𝑑 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
91	56, 90	jca 521	. . 3 ⊢ (𝜑 → (∃𝑐 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∃𝑑 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
92		reeanv 3234	. . 3 ⊢ (∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) ↔ (∃𝑐 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∃𝑑 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
93	91, 92	sylibr 237	. 2 ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
94		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
95	34, 69	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑖(∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
96	94, 95	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
97	1	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑋 ∈ Fin)
98	12	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑋 ≠ ∅)
99	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑌 ∈ (ℝ ↑_m 𝑋))
100		elmapi 8851	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℚ)
101		qssre 13011	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
102	101	a1i 11	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → ℚ ⊆ ℝ)
103	100, 102	fssd 6721	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℝ)
104	103	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) → 𝑐:𝑋⟶ℝ)
105	104	ad2antrr 739	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑐:𝑋⟶ℝ)
106		elmapi 8851	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℚ)
107	101	a1i 11	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → ℚ ⊆ ℝ)
108	106, 107	fssd 6721	. . . . . . . 8 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℝ)
109	108	ad2antlr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑:𝑋⟶ℝ)
110	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝐸 ∈ ℝ⁺)
111	35	elin2d 4151	. . . . . . . . 9 ⊢ ((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
112	111	adantlr 728	. . . . . . . 8 ⊢ (((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
113	112	adantll 727	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
114	70	elin2d 4151	. . . . . . . . 9 ⊢ ((∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
115	114	adantll 727	. . . . . . . 8 ⊢ (((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
116	115	adantll 727	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
117	96, 97, 98, 99, 105, 109, 110, 113, 116	hoiqssbllem1 47453	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)))
118		simpl 488	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → ((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)))
119		fveq2 6879	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑐‘𝑖) = (𝑐‘𝑘))
120		fveq2 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑘 → (𝑌‘𝑖) = (𝑌‘𝑘))
121	120	oveq1d 7429	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) = ((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋))))))
122	121, 120	oveq12d 7432	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)) = (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘)))
123	122	ineq2d 4166	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))))
124	119, 123	eleq12d 2854	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ↔ (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘)))))
125	124	cbvralvw 3240	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ↔ ∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))))
126	125	biimpi 219	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → ∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))))
127	126	adantr 486	. . . . . . . . 9 ⊢ ((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → ∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))))
128		fveq2 6879	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
129	120	oveq1d 7429	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = ((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
130	120, 129	oveq12d 7432	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))) = ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
131	130	ineq2d 4166	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
132	128, 131	eleq12d 2854	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → ((𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ↔ (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
133	132	cbvralvw 3240	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ↔ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
134	133	biimpi 219	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
135	134	adantl 487	. . . . . . . . 9 ⊢ ((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
136	127, 135	jca 521	. . . . . . . 8 ⊢ ((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
137	136	adantl 487	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
138		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
139	1	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑋 ∈ Fin)
140	12	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑋 ≠ ∅)
141	5	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑌 ∈ (ℝ ↑_m 𝑋))
142	104	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑐:𝑋⟶ℝ)
143	108	ad2antlr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑:𝑋⟶ℝ)
144	9	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝐸 ∈ ℝ⁺)
145	125, 111	sylanbr 594	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
146	145	adantlr 728	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
147	146	adantll 727	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
148	133, 114	sylanbr 594	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
149	148	adantll 727	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
150	149	adantll 727	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
151	138, 139, 140, 141, 142, 143, 144, 147, 150	hoiqssbllem2 47454	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑘 ∈ 𝑋 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))) ∧ ∀𝑘 ∈ 𝑋 (𝑑‘𝑘) ∈ (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
152	118, 137, 151	syl2anc 596	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
153	117, 152	jca 521	. . . . 5 ⊢ ((((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → (𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))
154	153	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → ((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → (𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))))
155	154	reximdva 3175	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) → (∃𝑑 ∈ (ℚ ↑_m 𝑋)(∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → ∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))))
156	155	reximdva 3175	. 2 ⊢ (𝜑 → (∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))))
157	93, 156	mpd 16	1 ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 class class class wbr 5103 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ↑_m cmap 8829 Xcixp 8907 Fincfn 8955 ℝcr 11126 + caddc 11130 · cmul 11132 < clt 11270 ≤ cle 11271 − cmin 11468 / cdiv 11898 ℕcn 12260 2c2 12322 ℚcq 13000 ℝ⁺crp 13045 (,)cioo 13401 [,)cico 13403 ♯chash 14397 √csqrt 15323 distcds 17354 ballcbl 21575 ℝ^crrx 25614

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13046 df-xadd 13167 df-ioo 13405 df-ico 13407 df-fz 13565 df-fzo 13713 df-seq 14069 df-exp 14129 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-clim 15578 df-sum 15777 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-starv 17360 df-sca 17361 df-vsca 17362 df-ip 17363 df-tset 17364 df-ple 17365 df-ds 17367 df-unif 17368 df-hom 17369 df-cco 17370 df-0g 17529 df-gsum 17530 df-prds 17535 df-pws 17537 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-mhm 18894 df-grp 19063 df-minusg 19064 df-sbg 19065 df-subg 19249 df-ghm 19344 df-cntz 19447 df-cmn 19912 df-abl 19913 df-mgp 20277 df-rng 20291 df-ur 20324 df-ring 20377 df-cring 20378 df-oppr 20481 df-dvdsr 20501 df-unit 20502 df-invr 20532 df-dvr 20545 df-rhm 20616 df-subrng 20711 df-subrg 20735 df-drng 20895 df-field 20896 df-staf 21008 df-srng 21009 df-lmod 21049 df-lss 21119 df-sra 21360 df-rgmod 21361 df-psmet 21580 df-xmet 21581 df-met 21582 df-bl 21583 df-cnfld 21589 df-refld 21821 df-dsmm 21948 df-frlm 21963 df-nm 24811 df-tng 24813 df-tcph 25400 df-rrx 25616

This theorem is used by: hoiqssbl 47456

W3C validator