hoiqssbllem2 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hoiqssbllem2		Structured version Visualization version GIF version

Theorem hoiqssbllem2 46638

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 2737	. . . . . . . . . 10 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
3		eqid 2737	. . . . . . . . . 10 ⊢ (ℝ ↑_m 𝑋) = (ℝ ↑_m 𝑋)
4	2, 3	rrxdsfi 25445	. . . . . . . . 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 480	. . . . . . 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 6905	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑌 → (𝑔‘𝑖) = (𝑌‘𝑖))
8	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (𝑔‘𝑖) = (𝑌‘𝑖))
9		fveq1 6905	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑓 → (ℎ‘𝑖) = (𝑓‘𝑖))
10	9	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (ℎ‘𝑖) = (𝑓‘𝑖))
11	8, 10	oveq12d 7449	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → ((𝑔‘𝑖) − (ℎ‘𝑖)) = ((𝑌‘𝑖) − (𝑓‘𝑖)))
12	11	oveq1d 7446	. . . . . . . . . 10 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
13	12	sumeq2sdv 15739	. . . . . . . . 9 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
14	13	fveq2d 6910	. . . . . . . 8 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (√‘Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2)) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)))
15	14	adantl 481	. . . . . . 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 480	. . . . . . 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	ffvelcdmda 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ)
21		hoiqssbllem2.d	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)
22	21	ffvelcdmda 7104	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ)
23	22	rexrd 11311	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ^*)
24	18, 20, 23	hoissrrn2 46593	. . . . . . . . 9 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (ℝ ↑_m 𝑋))
25	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (ℝ ↑_m 𝑋))
26		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
27	25, 26	sseldd 3984	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑_m 𝑋))
28		fvexd 6921	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ V)
29	6, 15, 17, 27, 28	ovmpod 7585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)))
30		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑓
31		nfixp1 8958	. . . . . . . . . 10 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))
32	30, 31	nfel 2920	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))
33	18, 32	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
34		simpl 482	. . . . . . . . 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 8889	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌:𝑋⟶ℝ)
37	16, 36	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
38	37	ffvelcdmda 7104	. . . . . . . . . . 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 13468	. . . . . . . . . . . . 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 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ ℝ)
43		fvixp2 45204	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
44	43	adantll 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
45	42, 44	sseldd 3984	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
46	39, 45	resubcld 11691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝑓‘𝑖)) ∈ ℝ)
47		2nn0 12543	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
48	47	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 2 ∈ ℕ₀)
49	46, 48	reexpcld 14203	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
50	33, 35, 49	fsumreclf 45591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
51		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
52		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
53	51, 52	oveq12d 7449	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
54	53	cbvixpv 8955	. . . . . . . . . . 11 ⊢ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))
55	54	eleq2i 2833	. . . . . . . . . 10 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
56	55	biimpi 216	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
57	56	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
58	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑋 ∈ Fin)
59		simpll 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝜑)
60	55	biimpri 228	. . . . . . . . . . 11 ⊢ (𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
61	60	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
62		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
63	59, 61, 62, 49	syl21anc 838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
64	46	sqge0d 14177	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
65	59, 61, 62, 64	syl21anc 838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
66	58, 63, 65	fsumge0 15831	. . . . . . . 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 15456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ ℝ)
69	29, 68	eqeltrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) ∈ ℝ)
70	22, 20	resubcld 11691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ)
71	70	resqcld 14165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ)
72	1, 71	fsumrecl 15770	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ)
73	70	sqge0d 14177	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
74	1, 71, 73	fsumge0 15831	. . . . . . 7 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
75	72, 74	resqrtcld 15456	. . . . . 6 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) ∈ ℝ)
76	75	adantr 480	. . . . 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 13077	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ)
79	78	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈ ℝ)
80		hoiqssbllem2.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ ∅)
81	80	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑋 ≠ ∅)
82	71	adantlr 715	. . . . . . . . 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 11691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ)
86	20	rexrd 11311	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ^*)
87	38	rexrd 11311	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
88		2rp 13039	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ∈ ℝ⁺
89	88	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 2 ∈ ℝ⁺)
90		hashnncl 14405	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
91	1, 90	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
92	80, 91	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
93	92	nnred 12281	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ)
94	92	nngt0d 12315	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 < (♯‘𝑋))
95	93, 94	elrpd 13074	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
96	95	rpsqrtcld 15450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
97	89, 96	rpmulcld 13093	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
98	77, 97	rpdivcld 13094	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
99	98	rpred 13077	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
100	99	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
101	38, 100	resubcld 11691	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
102	101	rexrd 11311	. . . . . . . . . . . . . . . . 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 45523	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝑌‘𝑖))
106	20, 38, 105	ltled 11409	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ≤ (𝑌‘𝑖))
107	38, 100	readdcld 11290	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
108	107	rexrd 11311	. . . . . . . . . . . . . . . 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 45508	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) < (𝐷‘𝑖))
112	86, 23, 87, 106, 111	elicod 13437	. . . . . . . . . . . . . 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 15474	. . . . . . . . . . . . 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 839	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑌‘𝑖) − (𝑓‘𝑖))) < ((𝐷‘𝑖) − (𝐶‘𝑖)))
116		0red 11264	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ∈ ℝ)
117	20, 38, 22, 106, 111	lelttrd 11419	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝐷‘𝑖))
118	20, 22	posdifd 11850	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐶‘𝑖) < (𝐷‘𝑖) ↔ 0 < ((𝐷‘𝑖) − (𝐶‘𝑖))))
119	117, 118	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 < ((𝐷‘𝑖) − (𝐶‘𝑖)))
120	116, 70, 119	ltled 11409	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐷‘𝑖) − (𝐶‘𝑖)))
121	70, 120	absidd 15461	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))) = ((𝐷‘𝑖) − (𝐶‘𝑖)))
122	121	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) = (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))))
123	122	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) = (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))))
124	115, 123	breqtrd 5169	. . . . . . . . . . 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 45371	. . . . . . . . . 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 838	. . . . . . . . 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 15836	. . . . . . . 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 15466	. . . . . . 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 232	. . . . . 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 5165	. . . . 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 13108	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈ ℝ)
135	134	resqcld 14165	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℝ)
136	135	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℝ)
137	22, 20	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ))
138	107, 101	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ))
139	137, 138	jca 511	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) ∈ ℝ ∧ (𝐶‘𝑖) ∈ ℝ) ∧ (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ ∧ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)))
140		iooltub 45523	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
142		ioogtlb 45508	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝐶‘𝑖))
144	141, 143	jca 511	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) < ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∧ ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) < (𝐶‘𝑖)))
145		lt2sub 11761	. . . . . . . . . . . 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 11289	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℂ)
148	100	recnd 11289	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℂ)
149	147, 148, 148	pnncand 11659	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))) = ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
150	78	recnd 11289	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ ℂ)
151	96	rpcnd 13079	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℂ)
152		2cnd 12344	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℂ)
153	96	rpne0d 13082	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (√‘(♯‘𝑋)) ≠ 0)
154	89	rpne0d 13082	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≠ 0)
155	150, 151, 152, 153, 154	divdiv3d 45370	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋))) / 2) = (𝐸 / (2 · (√‘(♯‘𝑋)))))
156	155	eqcomd 2743	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) = ((𝐸 / (√‘(♯‘𝑋))) / 2))
157	156, 156	oveq12d 7449	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2)))
158	150, 151, 153	divcld 12043	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈ ℂ)
159	158	2halvesd 12512	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2)) = (𝐸 / (√‘(♯‘𝑋))))
160	157, 159	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = (𝐸 / (√‘(♯‘𝑋))))
161	160	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = (𝐸 / (√‘(♯‘𝑋))))
162	149, 161	eqtrd 2777	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))) = (𝐸 / (√‘(♯‘𝑋))))
163	146, 162	breqtrd 5169	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋))))
164	134	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (√‘(♯‘𝑋))) ∈ ℝ)
165		0red 11264	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
166	96	rpred 13077	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ)
167	77	rpgt0d 13080	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝐸)
168	96	rpgt0d 13080	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (√‘(♯‘𝑋)))
169	78, 166, 167, 168	divgt0d 12203	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐸 / (√‘(♯‘𝑋))))
170	165, 134, 169	ltled 11409	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝐸 / (√‘(♯‘𝑋))))
171	170	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (𝐸 / (√‘(♯‘𝑋))))
172		lt2sq 14173	. . . . . . . . . . 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 839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋))) ↔ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2)))
174	163, 173	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < ((𝐸 / (√‘(♯‘𝑋)))↑2))
175	1, 80, 71, 136, 174	fsumlt 15836	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))
176	1, 136	fsumrecl 15770	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℝ)
177	164	sqge0d 14177	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐸 / (√‘(♯‘𝑋)))↑2))
178	1, 136, 177	fsumge0 15831	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))
179	72, 74, 176, 178	sqrtltd 15466	. . . . . . . 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 232	. . . . . . 7 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)))
181	135	recnd 11289	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℂ)
182		fsumconst 15826	. . . . . . . . . . 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 14161	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℂ ∧ (√‘(♯‘𝑋)) ∈ ℂ ∧ (√‘(♯‘𝑋)) ≠ 0) → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) / ((√‘(♯‘𝑋))↑2)))
185	150, 151, 153, 184	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) / ((√‘(♯‘𝑋))↑2)))
186	93	recnd 11289	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ)
187		sqrtth 15403	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑋) ∈ ℂ → ((√‘(♯‘𝑋))↑2) = (♯‘𝑋))
188	186, 187	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘(♯‘𝑋))↑2) = (♯‘𝑋))
189	188	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸↑2) / ((√‘(♯‘𝑋))↑2)) = ((𝐸↑2) / (♯‘𝑋)))
190	185, 189	eqtrd 2777	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) / (♯‘𝑋)))
191	190	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑋) · ((𝐸 / (√‘(♯‘𝑋)))↑2)) = ((♯‘𝑋) · ((𝐸↑2) / (♯‘𝑋))))
192	150	sqcld 14184	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸↑2) ∈ ℂ)
193	165, 94	gtned 11396	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑋) ≠ 0)
194	192, 186, 193	divcan2d 12045	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑋) · ((𝐸↑2) / (♯‘𝑋))) = (𝐸↑2))
195	183, 191, 194	3eqtrd 2781	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) = (𝐸↑2))
196	195	fveq2d 6910	. . . . . . . 8 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) = (√‘(𝐸↑2)))
197	165, 78, 167	ltled 11409	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝐸)
198		sqrtsq 15308	. . . . . . . . 9 ⊢ ((𝐸 ∈ ℝ ∧ 0 ≤ 𝐸) → (√‘(𝐸↑2)) = 𝐸)
199	78, 197, 198	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (√‘(𝐸↑2)) = 𝐸)
200		eqidd 2738	. . . . . . . 8 ⊢ (𝜑 → 𝐸 = 𝐸)
201	196, 199, 200	3eqtrd 2781	. . . . . . 7 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) = 𝐸)
202	180, 201	breqtrd 5169	. . . . . 6 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < 𝐸)
203	202	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2)) < 𝐸)
204	69, 76, 79, 133, 203	lttrd 11422	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) < 𝐸)
205		eqid 2737	. . . . . . . 8 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
206	205	rrxmetfi 25446	. . . . . . 7 ⊢ (𝑋 ∈ Fin → (dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ ↑_m 𝑋)))
207		metxmet 24344	. . . . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_m 𝑋)))
210	79	rexrd 11311	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈ ℝ^*)
211	27, 3	eleqtrdi 2851	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑_m 𝑋))
212		elbl2 24400	. . . . 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 839	. . . 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 257	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
215	214	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
216		dfss3 3972	. 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 234	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 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑_m cmap 8866 Xcixp 8937 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 + caddc 11158 · cmul 11160 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℝ⁺crp 13034 (,)cioo 13387 [,)cico 13389 ↑cexp 14102 ♯chash 14369 √csqrt 15272 abscabs 15273 Σcsu 15722 distcds 17306 ∞Metcxmet 21349 Metcmet 21350 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-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-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: hoiqssbllem3 46639

W3C validator