hoiqssbllem3 - Mathbox for Glauco Siliprandi

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Theorem hoiqssbllem3 44162

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 12701	. . . . . . . . 9 ⊢ ℚ ∈ V
3	2	inex1 5241	. . . . . . . 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 8637	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌:𝑋⟶ℝ)
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
8	7	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ)
9		hoiqssbllem3.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
10		2rp 12735	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ⁺
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ⁺)
12		hoiqssbllem3.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
13		hashnncl 14081	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
14	1, 13	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
15	12, 14	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
16		nnrp 12741	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑋) ∈ ℕ → (♯‘𝑋) ∈ ℝ⁺)
17	15, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
18	17	rpsqrtcld 15123	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
19	11, 18	rpmulcld 12788	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
20	9, 19	rpdivcld 12789	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
21	20	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
22	8, 21	ltsubrpd 12804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝑌‘𝑖))
23	21	rpred 12772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
24	8, 23	resubcld 11403	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
25	24, 8	ltnled 11122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝑌‘𝑖) ↔ ¬ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))))
26	22, 25	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ (𝑌‘𝑖) ≤ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))))
27	24	rexrd 11025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
28	8	rexrd 11025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
29	27, 28	qinioo 43073	. . . . . . . . 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 2950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) ≠ ∅)
32	1, 4, 31	choicefi 42740	. . . . . 6 ⊢ (𝜑 → ∃𝑐(𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))))
33		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))) → 𝑐 Fn 𝑋)
34		nfra1 3144	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
35		rspa 3132	. . . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝑖 ∈ 𝑋 → (𝑐‘𝑖) ∈ ℚ))
39	34, 38	ralrimi 3141	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ)
40	39	adantl 482	. . . . . . . . . . . . 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 512	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 6992	. . . . . . . . . . 11 ⊢ (𝑐:𝑋⟶ℚ ↔ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ ℚ))
44	42, 43	sylibr 233	. . . . . . . . . 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 8628	. . . . . . . . . . . 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 481	. . . . . . . . . 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 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))) → 𝑐 ∈ (ℚ ↑_m 𝑋))
50		simprr 770	. . . . . . . . 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 512	. . . . . . . 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 413	. . . . . . 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 1920	. . . . . 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 3070	. . . . 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 233	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑐‘𝑖) ∈ (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))))
57	2	inex1 5241	. . . . . . . 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 12805	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
60	8, 23	readdcld 11004	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
61	8, 60	ltnled 11122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ↔ ¬ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖)))
62	59, 61	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ¬ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ≤ (𝑌‘𝑖))
63	60	rexrd 11025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
64	28, 63	qinioo 43073	. . . . . . . . 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 2950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) ≠ ∅)
67	1, 58, 66	choicefi 42740	. . . . . 6 ⊢ (𝜑 → ∃𝑑(𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))))
68		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))) → 𝑑 Fn 𝑋)
69		nfra1 3144	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
70		rspa 3132	. . . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝑖 ∈ 𝑋 → (𝑑‘𝑖) ∈ ℚ))
74	69, 73	ralrimi 3141	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ)
75	74	adantl 482	. . . . . . . . . . . . 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 512	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 6992	. . . . . . . . . . 11 ⊢ (𝑑:𝑋⟶ℚ ↔ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ ℚ))
79	77, 78	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑:𝑋⟶ℚ)
80		elmapg 8628	. . . . . . . . . . . 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 481	. . . . . . . . . 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 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))) → 𝑑 ∈ (ℚ ↑_m 𝑋))
84		simprr 770	. . . . . . . . 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 512	. . . . . . . 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 413	. . . . . . 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 1920	. . . . . 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 3070	. . . . 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 233	. . . 4 ⊢ (𝜑 → ∃𝑑 ∈ (ℚ ↑_m 𝑋)∀𝑖 ∈ 𝑋 (𝑑‘𝑖) ∈ (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
91	56, 90	jca 512	. . 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 3294	. . 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 233	. 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 1917	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
95	34, 69	nfan 1902	. . . . . . . 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 1902	. . . . . . 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 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))))))))) → 𝑋 ∈ Fin)
98	12	ad3antrrr 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))))))))) → 𝑋 ≠ ∅)
99	5	ad3antrrr 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))))))))) → 𝑌 ∈ (ℝ ↑_m 𝑋))
100		elmapi 8637	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℚ)
101		qssre 12699	. . . . . . . . . . 11 ⊢ ℚ ⊆ ℝ
102	101	a1i 11	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → ℚ ⊆ ℝ)
103	100, 102	fssd 6618	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℝ)
104	103	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℚ ↑_m 𝑋)) → 𝑐:𝑋⟶ℝ)
105	104	ad2antrr 723	. . . . . . 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 8637	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℚ)
107	101	a1i 11	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → ℚ ⊆ ℝ)
108	106, 107	fssd 6618	. . . . . . . 8 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℝ)
109	108	ad2antlr 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))))))))) → 𝑑:𝑋⟶ℝ)
110	9	ad3antrrr 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))))))))) → 𝐸 ∈ ℝ⁺)
111	35	elin2d 4133	. . . . . . . . 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 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) → (𝑐‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
113	112	adantll 711	. . . . . . 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 4133	. . . . . . . . 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 711	. . . . . . . 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 711	. . . . . . 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 44160	. . . . . 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 483	. . . . . . 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 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑐‘𝑖) = (𝑐‘𝑘))
120		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑘 → (𝑌‘𝑖) = (𝑌‘𝑘))
121	120	oveq1d 7290	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) = ((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋))))))
122	121, 120	oveq12d 7293	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)) = (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘)))
123	122	ineq2d 4146	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (ℚ ∩ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) = (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))))
124	119, 123	eleq12d 2833	. . . . . . . . . . . 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 3383	. . . . . . . . . . 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 215	. . . . . . . . . 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 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 (𝑐‘𝑘) ∈ (ℚ ∩ (((𝑌‘𝑘) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑘))))
128		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
129	120	oveq1d 7290	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = ((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
130	120, 129	oveq12d 7293	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))) = ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
131	130	ineq2d 4146	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (ℚ ∩ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) = (ℚ ∩ ((𝑌‘𝑘)(,)((𝑌‘𝑘) + (𝐸 / (2 · (√‘(♯‘𝑋))))))))
132	128, 131	eleq12d 2833	. . . . . . . . . . . 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 3383	. . . . . . . . . . 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 215	. . . . . . . . . 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 482	. . . . . . . . 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 512	. . . . . . . 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 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))))))))) → (∀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 1917	. . . . . . . 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 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))))))))) → 𝑋 ∈ Fin)
140	12	ad3antrrr 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))))))))) → 𝑋 ≠ ∅)
141	5	ad3antrrr 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))))))))) → 𝑌 ∈ (ℝ ↑_m 𝑋))
142	104	ad2antrr 723	. . . . . . . 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 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))))))))) → 𝑑:𝑋⟶ℝ)
144	9	ad3antrrr 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))))))))) → 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 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) → (𝑐‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
147	146	adantll 711	. . . . . . . 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 711	. . . . . . . . 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 711	. . . . . . . 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 44161	. . . . . . 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 512	. . . . 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 413	. . . 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 3203	. . 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 3203	. 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 205 ∧ wa 396 = 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 Xcixp 8685 Fincfn 8733 ℝcr 10870 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 2c2 12028 ℚcq 12688 ℝ⁺crp 12730 (,)cioo 13079 [,)cico 13081 ♯chash 14044 √csqrt 14944 distcds 16971 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: hoiqssbl 44163

W3C validator