hoiqssbllem2 - Mathbox for Glauco Siliprandi

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Theorem hoiqssbllem2 42469

Description: The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
hoiqssbllem2.i	⊢ Ⅎ𝑖𝜑
hoiqssbllem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoiqssbllem2.n	⊢ (𝜑 → 𝑋 ≠ ∅)
hoiqssbllem2.y	⊢ (𝜑 → 𝑌 ∈ (ℝ ↑_𝑚 𝑋))
hoiqssbllem2.c	⊢ (𝜑 → 𝐶:𝑋⟶ℝ)
hoiqssbllem2.d	⊢ (𝜑 → 𝐷:𝑋⟶ℝ)
hoiqssbllem2.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
hoiqssbllem2.l	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
hoiqssbllem2.r	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))

**Assertion**
Ref	Expression
hoiqssbllem2	⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))

Distinct variable groups: 𝐶,𝑖 𝐷,𝑖 𝑖,𝐸 𝑖,𝑋 𝑖,𝑌 𝜑,𝑖

Proof of Theorem hoiqssbllem2 Dummy variables 𝑓 𝑔 ℎ 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoiqssbllem2.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		eqid 2797	. . . . . . . . . 10 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
3		eqid 2797	. . . . . . . . . 10 ⊢ (ℝ ↑_𝑚 𝑋) = (ℝ ↑_𝑚 𝑋)
4	2, 3	rrxdsfi 23701	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin → (dist‘(ℝ^‘𝑋)) = (𝑔 ∈ (ℝ ↑_𝑚 𝑋), ℎ ∈ (ℝ ↑_𝑚 𝑋) ↦ (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2))))
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) = (𝑔 ∈ (ℝ ↑_𝑚 𝑋), ℎ ∈ (ℝ ↑_𝑚 𝑋) ↦ (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2))))
6	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (dist‘(ℝ^‘𝑋)) = (𝑔 ∈ (ℝ ↑_𝑚 𝑋), ℎ ∈ (ℝ ↑_𝑚 𝑋) ↦ (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2))))
7		fveq1 6544	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑌 → (𝑔‘𝑖) = (𝑌‘𝑖))
8	7	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (𝑔‘𝑖) = (𝑌‘𝑖))
9		fveq1 6544	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑓 → (ℎ‘𝑖) = (𝑓‘𝑖))
10	9	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (ℎ‘𝑖) = (𝑓‘𝑖))
11	8, 10	oveq12d 7041	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → ((𝑔‘𝑖) − (ℎ‘𝑖)) = ((𝑌‘𝑖) − (𝑓‘𝑖)))
12	11	oveq1d 7038	. . . . . . . . . 10 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
13	12	sumeq2sdv 14898	. . . . . . . . 9 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
14	13	fveq2d 6549	. . . . . . . 8 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)))
15	14	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ (𝑔 = 𝑌 ∧ ℎ = 𝑓)) → (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)))
16		hoiqssbllem2.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑_𝑚 𝑋))
17	16	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑌 ∈ (ℝ ↑_𝑚 𝑋))
18		hoiqssbllem2.i	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝜑
19		hoiqssbllem2.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶:𝑋⟶ℝ)
20	19	ffvelrnda 6723	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ)
21		hoiqssbllem2.d	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)
22	21	ffvelrnda 6723	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ)
23	22	rexrd 10544	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ^*)
24	18, 20, 23	hoissrrn2 42424	. . . . . . . . 9 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (ℝ ↑_𝑚 𝑋))
25	24	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (ℝ ↑_𝑚 𝑋))
26		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
27	25, 26	sseldd 3896	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑_𝑚 𝑋))
28		fvexd 6560	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ V)
29	6, 15, 17, 27, 28	ovmpod 7165	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)))
30		nfcv 2951	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑓
31		nfixp1 8337	. . . . . . . . . 10 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))
32	30, 31	nfel 2963	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))
33	18, 32	nfan 1885	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
34		simpl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝜑)
35	34, 1	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑋 ∈ Fin)
36		elmapi 8285	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (ℝ ↑_𝑚 𝑋) → 𝑌:𝑋⟶ℝ)
37	16, 36	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
38	37	ffvelrnda 6723	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ)
39	34, 38	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ)
40		icossre 12671	. . . . . . . . . . . . 13 ⊢ (((𝐶‘𝑖) ∈ ℝ ∧ (𝐷‘𝑖) ∈ ℝ^*) → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ ℝ)
41	20, 23, 40	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ ℝ)
42	41	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ ℝ)
43		fvixp2 41022	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
44	43	adantll 710	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
45	42, 44	sseldd 3896	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
46	39, 45	resubcld 10922	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝑓‘𝑖)) ∈ ℝ)
47		2nn0 11768	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
48	47	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 2 ∈ ℕ₀)
49	46, 48	reexpcld 13381	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
50	33, 35, 49	fsumreclf 41420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
51		fveq2 6545	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
52		fveq2 6545	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
53	51, 52	oveq12d 7041	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
54	53	cbvixpv 8335	. . . . . . . . . . 11 ⊢ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))
55	54	eleq2i 2876	. . . . . . . . . 10 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
56	55	biimpi 217	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
57	56	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
58	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑋 ∈ Fin)
59		simpll 763	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝜑)
60	55	biimpri 229	. . . . . . . . . . 11 ⊢ (𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
61	60	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
62		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
63	59, 61, 62, 49	syl21anc 834	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
64	46	sqge0d 13466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
65	59, 61, 62, 64	syl21anc 834	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
66	58, 63, 65	fsumge0 14987	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
67	34, 57, 66	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
68	50, 67	resqrtcld 14615	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ ℝ)
69	29, 68	eqeltrd 2885	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) ∈ ℝ)
70	22, 20	resubcld 10922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ)
71	70	resqcld 13465	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ)
72	1, 71	fsumrecl 14928	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ)
73	70	sqge0d 13466	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
74	1, 71, 73	fsumge0 14987	. . . . . . 7 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
75	72, 74	resqrtcld 14615	. . . . . 6 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) ∈ ℝ)
76	75	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) ∈ ℝ)
77		hoiqssbllem2.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
78	77	rpred 12285	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ)
79	78	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈ ℝ)
80		hoiqssbllem2.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ ∅)
81	80	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑋 ≠ ∅)
82	71	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ)
83	34, 22	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ)
84	34, 20	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ)
85	83, 84	resubcld 10922	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ)
86	20	rexrd 10544	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ^*)
87	38	rexrd 10544	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
88		2rp 12248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ∈ ℝ⁺
89	88	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 2 ∈ ℝ⁺)
90		hashnncl 13581	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
91	1, 90	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
92	80, 91	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
93	92	nnred 11507	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ)
94	92	nngt0d 11540	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 < (♯‘𝑋))
95	93, 94	elrpd 12282	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
96	95	rpsqrtcld 14609	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
97	89, 96	rpmulcld 12301	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
98	77, 97	rpdivcld 12302	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
99	98	rpred 12285	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
100	99	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
101	38, 100	resubcld 10922	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
102	101	rexrd 10544	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
103		hoiqssbllem2.l	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖)))
104		iooltub 41349	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^* ∧ (𝑌‘𝑖) ∈ ℝ^* ∧ (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → (𝐶‘𝑖) < (𝑌‘𝑖))
105	102, 87, 103, 104	syl3anc 1364	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝑌‘𝑖))
106	20, 38, 105	ltled 10641	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ≤ (𝑌‘𝑖))
107	38, 100	readdcld 10523	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
108	107	rexrd 10544	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^*)
109		hoiqssbllem2.r	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋)))))))
110		ioogtlb 41333	. . . . . . . . . . . . . . . 16 ⊢ (((𝑌‘𝑖) ∈ ℝ^* ∧ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^* ∧ (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝑌‘𝑖) < (𝐷‘𝑖))
111	87, 108, 109, 110	syl3anc 1364	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < (𝐷‘𝑖))
112	86, 23, 87, 106, 111	elicod 12641	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
113	34, 112	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
114		icodiamlt 14633	. . . . . . . . . . . . 13 ⊢ ((((𝐶‘𝑖) ∈ ℝ ∧ (𝐷‘𝑖) ∈ ℝ) ∧ ((𝑌‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ∧ (𝑓‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))) → (abs‘((𝑌‘𝑖) − (𝑓‘𝑖))) < ((𝐷‘𝑖) − (𝐶‘𝑖)))
115	84, 83, 113, 44, 114	syl22anc 835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑌‘𝑖) − (𝑓‘𝑖))) < ((𝐷‘𝑖) − (𝐶‘𝑖)))
116		0red 10497	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ∈ ℝ)
117	20, 38, 22, 106, 111	lelttrd 10651	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝐷‘𝑖))
118	20, 22	posdifd 11081	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖) < (𝐷‘𝑖) ↔ 0 < ((𝐷‘𝑖) − (𝐶‘𝑖))))
119	117, 118	mpbid 233	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐷‘𝑖) − (𝐶‘𝑖)))
120	116, 70, 119	ltled 10641	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐷‘𝑖) − (𝐶‘𝑖)))
121	70, 120	absidd 14620	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))) = ((𝐷‘𝑖) − (𝐶‘𝑖)))
122	121	eqcomd 2803	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) = (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))))
123	122	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) = (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))))
124	115, 123	breqtrd 4994	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑌‘𝑖) − (𝑓‘𝑖))) < (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))))
125	46, 85, 124	abslt2sqd 41190	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
126	59, 61, 62, 125	syl21anc 834	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
127	58, 81, 63, 82, 126	fsumlt 14992	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
128	34, 57, 127	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
129	34, 72	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ)
130	34, 74	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
131	50, 67, 129, 130	sqrtltd 14625	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ↔ (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))))
132	128, 131	mpbid 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)))
133	29, 132	eqbrtrd 4990	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) < (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)))
134	78, 96	rerpdivcld 12316	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈ ℝ)
135	134	resqcld 13465	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℝ)
136	135	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℝ)
137	22, 20	jca 512	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ))
138	107, 101	jca 512	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ))
139	137, 138	jca 512	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ) ∧ (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)))
140		iooltub 41349	. . . . . . . . . . . . . 14 ⊢ (((𝑌‘𝑖) ∈ ℝ^* ∧ ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^* ∧ (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))) → (𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
141	87, 108, 109, 140	syl3anc 1364	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
142		ioogtlb 41333	. . . . . . . . . . . . . 14 ⊢ ((((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ^* ∧ (𝑌‘𝑖) ∈ ℝ^* ∧ (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))(,)(𝑌‘𝑖))) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝐶‘𝑖))
143	102, 87, 103, 142	syl3anc 1364	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝐶‘𝑖))
144	141, 143	jca 512	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∧ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝐶‘𝑖)))
145		lt2sub 10992	. . . . . . . . . . . 12 ⊢ ((((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ) ∧ (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)) → (((𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∧ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝐶‘𝑖)) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))))))
146	139, 144, 145	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))))
147	38	recnd 10522	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℂ)
148	100	recnd 10522	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℂ)
149	147, 148, 148	pnncand 10890	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))) = ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
150	78	recnd 10522	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ ℂ)
151	96	rpcnd 12287	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℂ)
152		2cnd 11569	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℂ)
153	96	rpne0d 12290	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (√‘(♯‘𝑋)) ≠ 0)
154	89	rpne0d 12290	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≠ 0)
155	150, 151, 152, 153, 154	divdiv3d 41189	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋))) / 2) = (𝐸 / (2 · (√‘(♯‘𝑋)))))
156	155	eqcomd 2803	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) = ((𝐸 / (√‘(♯‘𝑋))) / 2))
157	156, 156	oveq12d 7041	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2)))
158	150, 151, 153	divcld 11270	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈ ℂ)
159	158	2halvesd 11737	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2)) = (𝐸 / (√‘(♯‘𝑋))))
160	157, 159	eqtrd 2833	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = (𝐸 / (√‘(♯‘𝑋))))
161	160	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = (𝐸 / (√‘(♯‘𝑋))))
162	149, 161	eqtrd 2833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))) = (𝐸 / (√‘(♯‘𝑋))))
163	146, 162	breqtrd 4994	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋))))
164	134	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (√‘(♯‘𝑋))) ∈ ℝ)
165		0red 10497	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
166	96	rpred 12285	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ)
167	77	rpgt0d 12288	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝐸)
168	96	rpgt0d 12288	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (√‘(♯‘𝑋)))
169	78, 166, 167, 168	divgt0d 11429	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐸 / (√‘(♯‘𝑋))))
170	165, 134, 169	ltled 10641	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝐸 / (√‘(♯‘𝑋))))
171	170	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (𝐸 / (√‘(♯‘𝑋))))
172		lt2sq 13352	. . . . . . . . . . 11 ⊢ (((((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ ∧ 0 ≤ ((𝐷‘𝑖) − (𝐶‘𝑖))) ∧ ((𝐸 / (√‘(♯‘𝑋))) ∈ ℝ ∧ 0 ≤ (𝐸 / (√‘(♯‘𝑋))))) → (((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋))) ↔ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2)))
173	70, 120, 164, 171, 172	syl22anc 835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋))) ↔ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2)))
174	163, 173	mpbid 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2))
175	1, 80, 71, 136, 174	fsumlt 14992	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))
176	1, 136	fsumrecl 14928	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℝ)
177	164	sqge0d 13466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐸 / (√‘(♯‘𝑋)))↑2))
178	1, 136, 177	fsumge0 14987	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))
179	72, 74, 176, 178	sqrtltd 14625	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) ↔ (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))))
180	175, 179	mpbid 233	. . . . . . 7 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)))
181	135	recnd 10522	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℂ)
182		fsumconst 14982	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Fin ∧ ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℂ) → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((♯‘𝑋) · ((𝐸 / (√‘(♯‘𝑋)))↑2)))
183	1, 181, 182	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((♯‘𝑋) · ((𝐸 / (√‘(♯‘𝑋)))↑2)))
184		sqdiv 13341	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℂ ∧ (√‘(♯‘𝑋)) ∈ ℂ ∧ (√‘(♯‘𝑋)) ≠ 0) → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) / ((√‘(♯‘𝑋))↑2)))
185	150, 151, 153, 184	syl3anc 1364	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) / ((√‘(♯‘𝑋))↑2)))
186	93	recnd 10522	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ)
187		sqrtth 14562	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑋) ∈ ℂ → ((√‘(♯‘𝑋))↑2) = (♯‘𝑋))
188	186, 187	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘(♯‘𝑋))↑2) = (♯‘𝑋))
189	188	oveq2d 7039	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸↑2) / ((√‘(♯‘𝑋))↑2)) = ((𝐸↑2) / (♯‘𝑋)))
190	185, 189	eqtrd 2833	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) / (♯‘𝑋)))
191	190	oveq2d 7039	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑋) · ((𝐸 / (√‘(♯‘𝑋)))↑2)) = ((♯‘𝑋) · ((𝐸↑2) / (♯‘𝑋))))
192	150	sqcld 13362	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸↑2) ∈ ℂ)
193	165, 94	gtned 10628	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑋) ≠ 0)
194	192, 186, 193	divcan2d 11272	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑋) · ((𝐸↑2) / (♯‘𝑋))) = (𝐸↑2))
195	183, 191, 194	3eqtrd 2837	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) = (𝐸↑2))
196	195	fveq2d 6549	. . . . . . . 8 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) = (√‘(𝐸↑2)))
197	165, 78, 167	ltled 10641	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝐸)
198		sqrtsq 14467	. . . . . . . . 9 ⊢ ((𝐸 ∈ ℝ ∧ 0 ≤ 𝐸) → (√‘(𝐸↑2)) = 𝐸)
199	78, 197, 198	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (√‘(𝐸↑2)) = 𝐸)
200		eqidd 2798	. . . . . . . 8 ⊢ (𝜑 → 𝐸 = 𝐸)
201	196, 199, 200	3eqtrd 2837	. . . . . . 7 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) = 𝐸)
202	180, 201	breqtrd 4994	. . . . . 6 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < 𝐸)
203	202	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < 𝐸)
204	69, 76, 79, 133, 203	lttrd 10654	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) < 𝐸)
205		eqid 2797	. . . . . . . 8 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
206	205	rrxmetfi 23702	. . . . . . 7 ⊢ (𝑋 ∈ Fin → (dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ ↑_𝑚 𝑋)))
207		metxmet 22631	. . . . . . 7 ⊢ ((dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ ↑_𝑚 𝑋)) → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_𝑚 𝑋)))
208	1, 206, 207	3syl 18	. . . . . 6 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_𝑚 𝑋)))
209	208	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_𝑚 𝑋)))
210	79	rexrd 10544	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈ ℝ^*)
211	27, 3	syl6eleq 2895	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑_𝑚 𝑋))
212		elbl2 22687	. . . . 5 ⊢ ((((dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_𝑚 𝑋)) ∧ 𝐸 ∈ ℝ^*) ∧ (𝑌 ∈ (ℝ ↑_𝑚 𝑋) ∧ 𝑓 ∈ (ℝ ↑_𝑚 𝑋))) → (𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) ↔ (𝑌(dist‘(ℝ^‘𝑋))𝑓) < 𝐸))
213	209, 210, 17, 211, 212	syl22anc 835	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) ↔ (𝑌(dist‘(ℝ^‘𝑋))𝑓) < 𝐸))
214	204, 213	mpbird 258	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
215	214	ralrimiva 3151	. 2 ⊢ (𝜑 → ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
216		dfss3 3884	. 2 ⊢ (X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸) ↔ ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
217	215, 216	sylibr 235	1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 Ⅎwnf 1769 ∈ wcel 2083 ≠ wne 2986 ∀wral 3107 Vcvv 3440 ⊆ wss 3865 ∅c0 4217 class class class wbr 4968 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ∈ cmpo 7025 ↑_𝑚 cmap 8263 Xcixp 8317 Fincfn 8364 ℂcc 10388 ℝcr 10389 0cc0 10390 + caddc 10393 · cmul 10395 ℝ^*cxr 10527 < clt 10528 ≤ cle 10529 − cmin 10723 / cdiv 11151 ℕcn 11492 2c2 11546 ℕ₀cn0 11751 ℝ⁺crp 12243 (,)cioo 12592 [,)cico 12594 ↑cexp 13283 ♯chash 13544 √csqrt 14430 abscabs 14431 Σcsu 14880 distcds 16407 ∞Metcxmet 20216 Metcmet 20217 ballcbl 20218 ℝ^crrx 23673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-sup 8759 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-rp 12244 df-xadd 12362 df-ioo 12596 df-ico 12598 df-fz 12747 df-fzo 12888 df-seq 13224 df-exp 13284 df-hash 13545 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-clim 14683 df-sum 14881 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-hom 16422 df-cco 16423 df-0g 16548 df-gsum 16549 df-prds 16554 df-pws 16556 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-mhm 17778 df-grp 17868 df-minusg 17869 df-sbg 17870 df-subg 18034 df-ghm 18101 df-cntz 18192 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-cring 18994 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-dvr 19127 df-rnghom 19161 df-drng 19198 df-field 19199 df-subrg 19227 df-staf 19310 df-srng 19311 df-lmod 19330 df-lss 19398 df-sra 19638 df-rgmod 19639 df-psmet 20223 df-xmet 20224 df-met 20225 df-bl 20226 df-cnfld 20232 df-refld 20435 df-dsmm 20562 df-frlm 20577 df-nm 22879 df-tng 22881 df-tcph 23460 df-rrx 23675

This theorem is referenced by: hoiqssbllem3 42470

W3C validator