hoiqssbllem3 - Mathbox for Glauco Siliprandi

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Theorem hoiqssbllem3 42913

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 12363	. . . . . . . . 9 ⊢ ℚ ∈ V
3	2	inex1 5223	. . . . . . . 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 8430	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌:𝑋⟶ℝ)
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
8	7	ffvelrnda 6853	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ)
9		hoiqssbllem3.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
10		2rp 12397	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ⁺
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ⁺)
12		hoiqssbllem3.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
13		hashnncl 13730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
14	1, 13	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
15	12, 14	mpbird 259	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
16		nnrp 12403	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑋) ∈ ℕ → (♯‘𝑋) ∈ ℝ⁺)
17	15, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
18	17	rpsqrtcld 14773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
19	11, 18	rpmulcld 12450	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
20	9, 19	rpdivcld 12451	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
21	20	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
22	8, 21	ltsubrpd 12466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝑌‘𝑖))
23	21	rpred 12434	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
24	8, 23	resubcld 11070	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
25	24, 8	ltnled 10789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝑌‘𝑖) ↔ ¬ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))))
26	22, 25	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))))
27	24	rexrd 10693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
28	8	rexrd 10693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
29	27, 28	qinioo 41818	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = ∅ ↔ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))))
30	26, 29	mtbird 327	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = ∅)
31	30	neqned 3025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ≠ ∅)
32	1, 4, 31	choicefi 41470	. . . . . 6 ⊢ (𝜑 → ∃𝑐(𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))))
33		simpl 485	. . . . . . . . . . . . 13 ⊢ ((𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))) → 𝑐 Fn 𝑋)
34		nfra1 3221	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
35		rspa 3208	. . . . . . . . . . . . . . . . 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 4174	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝑖 ∈ 𝑋 → (𝑐‘𝑖) ∈ ℚ))
39	34, 38	ralrimi 3218	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ)
40	39	adantl 484	. . . . . . . . . . . . 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 514	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 6884	. . . . . . . . . . 11 ⊢ (𝑐:𝑋⟶ℚ ↔ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ))
44	42, 43	sylibr 236	. . . . . . . . . 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 8421	. . . . . . . . . . . 12 ⊢ ((ℚ ∈ V ∧ 𝑋 ∈ Fin) → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐:𝑋⟶ℚ))
47	45, 1, 46	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ (ℚ ↑_m 𝑋) ↔ 𝑐:𝑋⟶ℚ))
48	47	adantr 483	. . . . . . . . . 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 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → 𝑐 ∈ (ℚ ↑_m 𝑋))
50		simprr 771	. . . . . . . . 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 514	. . . . . . . 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 415	. . . . . . 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 1918	. . . . . 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 3146	. . . . 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 236	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))
57	2	inex1 5223	. . . . . . . 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 12467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
60	8, 23	readdcld 10672	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
61	8, 60	ltnled 10789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ↔ ¬ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖)))
62	59, 61	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖))
63	60	rexrd 10693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
64	28, 63	qinioo 41818	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = ∅ ↔ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖)))
65	62, 64	mtbird 327	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = ∅)
66	65	neqned 3025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ≠ ∅)
67	1, 58, 66	choicefi 41470	. . . . . 6 ⊢ (𝜑 → ∃𝑑(𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
68		simpl 485	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → 𝑑 Fn 𝑋)
69		nfra1 3221	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
70		rspa 3208	. . . . . . . . . . . . . . . . 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 4174	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝑖 ∈ 𝑋 → (𝑑‘𝑖) ∈ ℚ))
74	69, 73	ralrimi 3218	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ)
75	74	adantl 484	. . . . . . . . . . . . 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 514	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 6884	. . . . . . . . . . 11 ⊢ (𝑑:𝑋⟶ℚ ↔ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ))
79	77, 78	sylibr 236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑:𝑋⟶ℚ)
80		elmapg 8421	. . . . . . . . . . . 12 ⊢ ((ℚ ∈ V ∧ 𝑋 ∈ Fin) → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑:𝑋⟶ℚ))
81	45, 1, 80	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (ℚ ↑_m 𝑋) ↔ 𝑑:𝑋⟶ℚ))
82	81	adantr 483	. . . . . . . . . 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 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑 ∈ (ℚ ↑_m 𝑋))
84		simprr 771	. . . . . . . . 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 514	. . . . . . . 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 415	. . . . . . 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 1918	. . . . . 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 3146	. . . . 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 236	. . . 4 ⊢ (𝜑 → ∃𝑑 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
91	56, 90	jca 514	. . 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 3369	. . 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 236	. 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 1915	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
95	34, 69	nfan 1900	. . . . . . . 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 1900	. . . . . . 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 728	. . . . . . 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 728	. . . . . . 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 728	. . . . . . 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 8430	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℚ)
101		qssre 12361	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
102	101	a1i 11	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → ℚ ⊆ ℝ)
103	100, 102	fssd 6530	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℝ)
104	103	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) → 𝑐:𝑋⟶ℝ)
105	104	ad2antrr 724	. . . . . . 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 8430	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℚ)
107	101	a1i 11	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → ℚ ⊆ ℝ)
108	106, 107	fssd 6530	. . . . . . . 8 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℝ)
109	108	ad2antlr 725	. . . . . . 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 728	. . . . . . 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 4178	. . . . . . . . 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 713	. . . . . . . 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 712	. . . . . . 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 4178	. . . . . . . . 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 712	. . . . . . . 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 712	. . . . . . 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 42911	. . . . . 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 485	. . . . . . 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 6672	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑐‘𝑖) = (𝑐‘𝑘))
120		fveq2 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑘 → (𝑌‘𝑖) = (𝑌‘𝑘))
121	120	oveq1d 7173	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) = ((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋))))))
122	121, 120	oveq12d 7176	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)) = (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘)))
123	122	ineq2d 4191	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))))
124	119, 123	eleq12d 2909	. . . . . . . . . . . 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 3451	. . . . . . . . . . 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 218	. . . . . . . . . 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 483	. . . . . . . . 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 6672	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
129	120	oveq1d 7173	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = ((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
130	120, 129	oveq12d 7176	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))) = ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
131	130	ineq2d 4191	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
132	128, 131	eleq12d 2909	. . . . . . . . . . . 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 3451	. . . . . . . . . . 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 218	. . . . . . . . . 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 484	. . . . . . . . 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 514	. . . . . . . 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 484	. . . . . . 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 1915	. . . . . . . 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 728	. . . . . . . 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 728	. . . . . . . 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 728	. . . . . . . 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 724	. . . . . . . 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 725	. . . . . . . 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 728	. . . . . . . 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 584	. . . . . . . . . 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 713	. . . . . . . . 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 712	. . . . . . . 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 584	. . . . . . . . . 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 712	. . . . . . . . 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 712	. . . . . . . 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 42912	. . . . . . 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 586	. . . . . 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 514	. . . . 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 415	. . . 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 3276	. . 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 3276	. 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 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑_m cmap 8408 Xcixp 8463 Fincfn 8511 ℝcr 10538 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 2c2 11695 ℚcq 12351 ℝ⁺crp 12392 (,)cioo 12741 [,)cico 12743 ♯chash 13693 √csqrt 14594 distcds 16576 ballcbl 20534 ℝ^crrx 23988

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xadd 12511 df-ioo 12745 df-ico 12747 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-rnghom 19469 df-drng 19506 df-field 19507 df-subrg 19535 df-staf 19618 df-srng 19619 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-cnfld 20548 df-refld 20751 df-dsmm 20878 df-frlm 20893 df-nm 23194 df-tng 23196 df-tcph 23775 df-rrx 23990

This theorem is referenced by: hoiqssbl 42914

W3C validator