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 46877

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 2736	. . . . . . . . . 10 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
3		eqid 2736	. . . . . . . . . 10 ⊢ (ℝ ↑_m 𝑋) = (ℝ ↑_m 𝑋)
4	2, 3	rrxdsfi 25367	. . . . . . . . 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 6833	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑌 → (𝑔‘𝑖) = (𝑌‘𝑖))
8	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (𝑔‘𝑖) = (𝑌‘𝑖))
9		fveq1 6833	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑓 → (ℎ‘𝑖) = (𝑓‘𝑖))
10	9	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (ℎ‘𝑖) = (𝑓‘𝑖))
11	8, 10	oveq12d 7376	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → ((𝑔‘𝑖) − (ℎ‘𝑖)) = ((𝑌‘𝑖) − (𝑓‘𝑖)))
12	11	oveq1d 7373	. . . . . . . . . 10 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
13	12	sumeq2sdv 15626	. . . . . . . . 9 ⊢ ((𝑔 = 𝑌 ∧ ℎ = 𝑓) → Σ𝑖 ∈ 𝑋 (((𝑔‘𝑖) − (ℎ‘𝑖))↑2) = Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
14	13	fveq2d 6838	. . . . . . . 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 7029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ)
21		hoiqssbllem2.d	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)
22	21	ffvelcdmda 7029	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ)
23	22	rexrd 11182	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℝ^*)
24	18, 20, 23	hoissrrn2 46832	. . . . . . . . 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 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑_m 𝑋))
28		fvexd 6849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ V)
29	6, 15, 17, 27, 28	ovmpod 7510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) = (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)))
30		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑓
31		nfixp1 8856	. . . . . . . . . 10 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))
32	30, 31	nfel 2913	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))
33	18, 32	nfan 1900	. . . . . . . 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 8786	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (ℝ ↑_m 𝑋) → 𝑌:𝑋⟶ℝ)
37	16, 36	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌:𝑋⟶ℝ)
38	37	ffvelcdmda 7029	. . . . . . . . . . 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 13344	. . . . . . . . . . . . 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 45453	. . . . . . . . . . . 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 3934	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
46	39, 45	resubcld 11565	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝑓‘𝑖)) ∈ ℝ)
47		2nn0 12418	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
48	47	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 2 ∈ ℕ₀)
49	46, 48	reexpcld 14086	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
50	33, 35, 49	fsumreclf 45832	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
51		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
52		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
53	51, 52	oveq12d 7376	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
54	53	cbvixpv 8853	. . . . . . . . . . 11 ⊢ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))
55	54	eleq2i 2828	. . . . . . . . . 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 766	. . . . . . . . . 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 837	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) − (𝑓‘𝑖))↑2) ∈ ℝ)
64	46	sqge0d 14060	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
65	59, 61, 62, 64	syl21anc 837	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝑌‘𝑖) − (𝑓‘𝑖))↑2))
66	58, 63, 65	fsumge0 15718	. . . . . . . 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 15341	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (√‘Σ𝑖 ∈ 𝑋 (((𝑌‘𝑖) − (𝑓‘𝑖))↑2)) ∈ ℝ)
69	29, 68	eqeltrd 2836	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) ∈ ℝ)
70	22, 20	resubcld 11565	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ)
71	70	resqcld 14048	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ)
72	1, 71	fsumrecl 15657	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) ∈ ℝ)
73	70	sqge0d 14060	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
74	1, 71, 73	fsumge0 15718	. . . . . . 7 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2))
75	72, 74	resqrtcld 15341	. . . . . 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 12949	. . . . . 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 11565	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) ∈ ℝ)
86	20	rexrd 11182	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℝ^*)
87	38	rexrd 11182	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℝ^*)
88		2rp 12910	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ∈ ℝ⁺
89	88	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 2 ∈ ℝ⁺)
90		hashnncl 14289	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12160	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ)
94	92	nngt0d 12194	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 < (♯‘𝑋))
95	93, 94	elrpd 12946	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (♯‘𝑋) ∈ ℝ⁺)
96	95	rpsqrtcld 15335	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ⁺)
97	89, 96	rpmulcld 12965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (2 · (√‘(♯‘𝑋))) ∈ ℝ⁺)
98	77, 97	rpdivcld 12966	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ⁺)
99	98	rpred 12949	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
100	99	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℝ)
101	38, 100	resubcld 11565	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
102	101	rexrd 11182	. . . . . . . . . . . . . . . . 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 45766	. . . . . . . . . . . . . . . . 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 11281	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ≤ (𝑌‘𝑖))
107	38, 100	readdcld 11161	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) ∈ ℝ)
108	107	rexrd 11182	. . . . . . . . . . . . . . . 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 45751	. . . . . . . . . . . . . . . 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 13311	. . . . . . . . . . . . . 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 15361	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) ∧ 𝑖 ∈ 𝑋) → (abs‘((𝑌‘𝑖) − (𝑓‘𝑖))) < ((𝐷‘𝑖) − (𝐶‘𝑖)))
116		0red 11135	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ∈ ℝ)
117	20, 38, 22, 106, 111	lelttrd 11291	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) < (𝐷‘𝑖))
118	20, 22	posdifd 11724	. . . . . . . . . . . . . . . . 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 11281	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐷‘𝑖) − (𝐶‘𝑖)))
121	70, 120	absidd 15346	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (abs‘((𝐷‘𝑖) − (𝐶‘𝑖))) = ((𝐷‘𝑖) − (𝐶‘𝑖)))
122	121	eqcomd 2742	. . . . . . . . . . . . 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 5124	. . . . . . . . . . 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 45615	. . . . . . . . . 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 837	. . . . . . . . 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 15723	. . . . . . . 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 15351	. . . . . . 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 5120	. . . . 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 12980	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈ ℝ)
135	134	resqcld 14048	. . . . . . . . . 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 45766	. . . . . . . . . . . . . 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 45751	. . . . . . . . . . . . . 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 11635	. . . . . . . . . . . 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 11160	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑌‘𝑖) ∈ ℂ)
148	100	recnd 11160	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (2 · (√‘(♯‘𝑋)))) ∈ ℂ)
149	147, 148, 148	pnncand 11531	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))) = ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))))
150	78	recnd 11160	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ ℂ)
151	96	rpcnd 12951	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℂ)
152		2cnd 12223	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℂ)
153	96	rpne0d 12954	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (√‘(♯‘𝑋)) ≠ 0)
154	89	rpne0d 12954	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≠ 0)
155	150, 151, 152, 153, 154	divdiv3d 45614	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋))) / 2) = (𝐸 / (2 · (√‘(♯‘𝑋)))))
156	155	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 / (2 · (√‘(♯‘𝑋)))) = ((𝐸 / (√‘(♯‘𝑋))) / 2))
157	156, 156	oveq12d 7376	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸 / (2 · (√‘(♯‘𝑋)))) + (𝐸 / (2 · (√‘(♯‘𝑋))))) = (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2)))
158	150, 151, 153	divcld 11917	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 / (√‘(♯‘𝑋))) ∈ ℂ)
159	158	2halvesd 12387	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐸 / (√‘(♯‘𝑋))) / 2) + ((𝐸 / (√‘(♯‘𝑋))) / 2)) = (𝐸 / (√‘(♯‘𝑋))))
160	157, 159	eqtrd 2771	. . . . . . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (((𝑌‘𝑖) + (𝐸 / (2 · (√‘(♯‘𝑋))))) − ((𝑌‘𝑖) − (𝐸 / (2 · (√‘(♯‘𝑋)))))) = (𝐸 / (√‘(♯‘𝑋))))
163	146, 162	breqtrd 5124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐷‘𝑖) − (𝐶‘𝑖)) < (𝐸 / (√‘(♯‘𝑋))))
164	134	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐸 / (√‘(♯‘𝑋))) ∈ ℝ)
165		0red 11135	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
166	96	rpred 12949	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (√‘(♯‘𝑋)) ∈ ℝ)
167	77	rpgt0d 12952	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝐸)
168	96	rpgt0d 12952	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (√‘(♯‘𝑋)))
169	78, 166, 167, 168	divgt0d 12077	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (𝐸 / (√‘(♯‘𝑋))))
170	165, 134, 169	ltled 11281	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝐸 / (√‘(♯‘𝑋))))
171	170	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ (𝐸 / (√‘(♯‘𝑋))))
172		lt2sq 14056	. . . . . . . . . . 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 838	. . . . . . . . . 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 15723	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 (((𝐷‘𝑖) − (𝐶‘𝑖))↑2) < Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))
176	1, 136	fsumrecl 15657	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℝ)
177	164	sqge0d 14060	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 0 ≤ ((𝐸 / (√‘(♯‘𝑋)))↑2))
178	1, 136, 177	fsumge0 15718	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2))
179	72, 74, 176, 178	sqrtltd 15351	. . . . . . . 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 11160	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) ∈ ℂ)
182		fsumconst 15713	. . . . . . . . . . 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 14044	. . . . . . . . . . . . 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 11160	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ)
187		sqrtth 15288	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑋) ∈ ℂ → ((√‘(♯‘𝑋))↑2) = (♯‘𝑋))
188	186, 187	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘(♯‘𝑋))↑2) = (♯‘𝑋))
189	188	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸↑2) / ((√‘(♯‘𝑋))↑2)) = ((𝐸↑2) / (♯‘𝑋)))
190	185, 189	eqtrd 2771	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / (√‘(♯‘𝑋)))↑2) = ((𝐸↑2) / (♯‘𝑋)))
191	190	oveq2d 7374	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑋) · ((𝐸 / (√‘(♯‘𝑋)))↑2)) = ((♯‘𝑋) · ((𝐸↑2) / (♯‘𝑋))))
192	150	sqcld 14067	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸↑2) ∈ ℂ)
193	165, 94	gtned 11268	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑋) ≠ 0)
194	192, 186, 193	divcan2d 11919	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑋) · ((𝐸↑2) / (♯‘𝑋))) = (𝐸↑2))
195	183, 191, 194	3eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2) = (𝐸↑2))
196	195	fveq2d 6838	. . . . . . . 8 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) = (√‘(𝐸↑2)))
197	165, 78, 167	ltled 11281	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝐸)
198		sqrtsq 15192	. . . . . . . . 9 ⊢ ((𝐸 ∈ ℝ ∧ 0 ≤ 𝐸) → (√‘(𝐸↑2)) = 𝐸)
199	78, 197, 198	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (√‘(𝐸↑2)) = 𝐸)
200		eqidd 2737	. . . . . . . 8 ⊢ (𝜑 → 𝐸 = 𝐸)
201	196, 199, 200	3eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → (√‘Σ𝑖 ∈ 𝑋 ((𝐸 / (√‘(♯‘𝑋)))↑2)) = 𝐸)
202	180, 201	breqtrd 5124	. . . . . 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 11294	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → (𝑌(dist‘(ℝ^‘𝑋))𝑓) < 𝐸)
205		eqid 2736	. . . . . . . 8 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
206	205	rrxmetfi 25368	. . . . . . 7 ⊢ (𝑋 ∈ Fin → (dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ ↑_m 𝑋)))
207		metxmet 24278	. . . . . . 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 11182	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝐸 ∈ ℝ^*)
211	27, 3	eleqtrdi 2846	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) → 𝑓 ∈ (ℝ ↑_m 𝑋))
212		elbl2 24334	. . . . 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 838	. . . 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 3128	. 2 ⊢ (𝜑 → ∀𝑓 ∈ X 𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))𝑓 ∈ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
216		dfss3 3922	. 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 1541 Ⅎwnf 1784 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 Vcvv 3440 ⊆ wss 3901 ∅c0 4285 class class class wbr 5098 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ↑_m cmap 8763 Xcixp 8835 Fincfn 8883 ℂcc 11024 ℝcr 11025 0cc0 11026 + caddc 11029 · cmul 11031 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 / cdiv 11794 ℕcn 12145 2c2 12200 ℕ₀cn0 12401 ℝ⁺crp 12905 (,)cioo 13261 [,)cico 13263 ↑cexp 13984 ♯chash 14253 √csqrt 15156 abscabs 15157 Σcsu 15609 distcds 17186 ∞Metcxmet 21294 Metcmet 21295 ballcbl 21296 ℝ^crrx 25339

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-xadd 13027 df-ioo 13265 df-ico 13267 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-0g 17361 df-gsum 17362 df-prds 17367 df-pws 17369 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-ghm 19142 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-rhm 20408 df-subrng 20479 df-subrg 20503 df-drng 20664 df-field 20665 df-staf 20772 df-srng 20773 df-lmod 20813 df-lss 20883 df-sra 21125 df-rgmod 21126 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-cnfld 21310 df-refld 21560 df-dsmm 21687 df-frlm 21702 df-nm 24526 df-tng 24528 df-tcph 25125 df-rrx 25341

This theorem is referenced by: hoiqssbllem3 46878

W3C validator