hoiqssbllem3 - Mathbox for Glauco Siliprandi

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Theorem hoiqssbllem3 46639

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 13003	. . . . . . . . 9 ⊢ ℚ ∈ V
3	2	inex1 5317	. . . . . . . 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 8889	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌:𝑋⟶ℝ)
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
8	7	ffvelcdmda 7104	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ)
9		hoiqssbllem3.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
10		2rp 13039	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ⁺
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ⁺)
12		hoiqssbllem3.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
13		hashnncl 14405	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
14	1, 13	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
15	12, 14	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
16		nnrp 13046	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑋) ∈ ℕ → (♯‘𝑋) ∈ ℝ⁺)
17	15, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
18	17	rpsqrtcld 15450	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
19	11, 18	rpmulcld 13093	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
20	9, 19	rpdivcld 13094	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
21	20	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
22	8, 21	ltsubrpd 13109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝑌‘𝑖))
23	21	rpred 13077	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
24	8, 23	resubcld 11691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
25	24, 8	ltnled 11408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝑌‘𝑖) ↔ ¬ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))))
26	22, 25	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))))
27	24	rexrd 11311	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
28	8	rexrd 11311	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
29	27, 28	qinioo 45548	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = ∅ ↔ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))))
30	26, 29	mtbird 325	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = ∅)
31	30	neqned 2947	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ≠ ∅)
32	1, 4, 31	choicefi 45205	. . . . . 6 ⊢ (𝜑 → ∃𝑐(𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))))
33		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))) → 𝑐 Fn 𝑋)
34		nfra1 3284	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
35		rspa 3248	. . . . . . . . . . . . . . . . 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 4201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝑐‘𝑖) ∈ ℚ)
37	35, 36	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑐‘𝑖) ∈ ℚ)
38	37	ex 412	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝑖 ∈ 𝑋 → (𝑐‘𝑖) ∈ ℚ))
39	34, 38	ralrimi 3257	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ)
40	39	adantl 481	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 7139	. . . . . . . . . . 11 ⊢ (𝑐:𝑋⟶ℚ ↔ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ))
44	42, 43	sylibr 234	. . . . . . . . . 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 8879	. . . . . . . . . . . 12 ⊢ ((ℚ ∈ V ∧ 𝑋 ∈ Fin) → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐:𝑋⟶ℚ))
47	45, 1, 46	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐:𝑋⟶ℚ))
48	47	adantr 480	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → 𝑐 ∈ (ℚ ↑_m 𝑋))
50		simprr 773	. . . . . . . . 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 511	. . . . . . . 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 412	. . . . . . 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 1917	. . . . . 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 15	. . . . 5 ⊢ (𝜑 → ∃𝑐(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))))
55		df-rex 3071	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))
57	2	inex1 5317	. . . . . . . 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 13110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
60	8, 23	readdcld 11290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
61	8, 60	ltnled 11408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ↔ ¬ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖)))
62	59, 61	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖))
63	60	rexrd 11311	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
64	28, 63	qinioo 45548	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = ∅ ↔ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖)))
65	62, 64	mtbird 325	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = ∅)
66	65	neqned 2947	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ≠ ∅)
67	1, 58, 66	choicefi 45205	. . . . . 6 ⊢ (𝜑 → ∃𝑑(𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
68		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → 𝑑 Fn 𝑋)
69		nfra1 3284	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
70		rspa 3248	. . . . . . . . . . . . . . . . 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 4201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝑑‘𝑖) ∈ ℚ)
72	70, 71	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ∧ 𝑖 ∈ 𝑋) → (𝑑‘𝑖) ∈ ℚ)
73	72	ex 412	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝑖 ∈ 𝑋 → (𝑑‘𝑖) ∈ ℚ))
74	69, 73	ralrimi 3257	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ)
75	74	adantl 481	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 7139	. . . . . . . . . . 11 ⊢ (𝑑:𝑋⟶ℚ ↔ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ))
79	77, 78	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑:𝑋⟶ℚ)
80		elmapg 8879	. . . . . . . . . . . 12 ⊢ ((ℚ ∈ V ∧ 𝑋 ∈ Fin) → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑:𝑋⟶ℚ))
81	45, 1, 80	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑:𝑋⟶ℚ))
82	81	adantr 480	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑 ∈ (ℚ ↑_m 𝑋))
84		simprr 773	. . . . . . . . 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 511	. . . . . . . 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 412	. . . . . . 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 1917	. . . . . 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 15	. . . . 5 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ (ℚ ↑_m 𝑋) ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
89		df-rex 3071	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∃𝑑 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
91	56, 90	jca 511	. . 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 3229	. . 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 234	. 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 1914	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
95	34, 69	nfan 1899	. . . . . . . 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 1899	. . . . . . 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 730	. . . . . . 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 730	. . . . . . 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 730	. . . . . . 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 8889	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℚ)
101		qssre 13001	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
102	101	a1i 11	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → ℚ ⊆ ℝ)
103	100, 102	fssd 6753	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℝ)
104	103	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) → 𝑐:𝑋⟶ℝ)
105	104	ad2antrr 726	. . . . . . 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 8889	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℚ)
107	101	a1i 11	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → ℚ ⊆ ℝ)
108	106, 107	fssd 6753	. . . . . . . 8 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℝ)
109	108	ad2antlr 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))))))))) → 𝑑:𝑋⟶ℝ)
110	9	ad3antrrr 730	. . . . . . 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 4205	. . . . . . . . 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 715	. . . . . . . 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 714	. . . . . . 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 4205	. . . . . . . . 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 714	. . . . . . . 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 714	. . . . . . 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 46637	. . . . . 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 482	. . . . . . 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 6906	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑐‘𝑖) = (𝑐‘𝑘))
120		fveq2 6906	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑘 → (𝑌‘𝑖) = (𝑌‘𝑘))
121	120	oveq1d 7446	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) = ((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋))))))
122	121, 120	oveq12d 7449	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)) = (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘)))
123	122	ineq2d 4220	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))))
124	119, 123	eleq12d 2835	. . . . . . . . . . . 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 3237	. . . . . . . . . . 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 216	. . . . . . . . . 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 480	. . . . . . . . 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 6906	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
129	120	oveq1d 7446	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = ((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
130	120, 129	oveq12d 7449	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))) = ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
131	130	ineq2d 4220	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
132	128, 131	eleq12d 2835	. . . . . . . . . . . 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 3237	. . . . . . . . . . 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 216	. . . . . . . . . 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 481	. . . . . . . . 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 511	. . . . . . . 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 481	. . . . . . 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 1914	. . . . . . . 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 730	. . . . . . . 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 730	. . . . . . . 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 730	. . . . . . . 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 726	. . . . . . . 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 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))))))))) → 𝑑:𝑋⟶ℝ)
144	9	ad3antrrr 730	. . . . . . . 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 582	. . . . . . . . . 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 715	. . . . . . . . 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 714	. . . . . . . 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 582	. . . . . . . . . 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 714	. . . . . . . . 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 714	. . . . . . . 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 46638	. . . . . . 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 584	. . . . . 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 511	. . . . 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 412	. . . 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 3168	. . 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 3168	. 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 15	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = 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 Xcixp 8937 Fincfn 8985 ℝcr 11154 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℚcq 12990 ℝ⁺crp 13034 (,)cioo 13387 [,)cico 13389 ♯chash 14369 √csqrt 15272 distcds 17306 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: hoiqssbl 46640

W3C validator