rrndstprj2 - Mathbox for Jeff Madsen

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Theorem rrndstprj2 37212

Description: Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 37211 can be used to show that the supremum norm and Euclidean norm are equivalent. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

**Hypotheses**
Ref	Expression
rrnval.1	⊢ 𝑋 = (ℝ ↑_m 𝐼)
rrndstprj1.1	⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))

**Assertion**
Ref	Expression
rrndstprj2	⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝐹(ℝⁿ‘𝐼)𝐺) < (𝑅 · (√‘(♯‘𝐼))))

Distinct variable groups: 𝑛,𝐺 𝑛,𝐼 𝑛,𝑀 𝑅,𝑛 𝑛,𝐹 Allowed substitution hint: 𝑋(𝑛)

Proof of Theorem rrndstprj2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1188	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐼 ∈ (Fin ∖ {∅}))
2	1	eldifad 3955	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐼 ∈ Fin)
3		simpl2 1189	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐹 ∈ 𝑋)
4		simpl3 1190	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐺 ∈ 𝑋)
5		rrnval.1	. . . 4 ⊢ 𝑋 = (ℝ ↑_m 𝐼)
6	5	rrnmval 37209	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(ℝⁿ‘𝐼)𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
7	2, 3, 4, 6	syl3anc 1368	. 2 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝐹(ℝⁿ‘𝐼)𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
8		eldifsni 4788	. . . . . 6 ⊢ (𝐼 ∈ (Fin ∖ {∅}) → 𝐼 ≠ ∅)
9	1, 8	syl 17	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐼 ≠ ∅)
10	3, 5	eleqtrdi 2837	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐹 ∈ (ℝ ↑_m 𝐼))
11		elmapi 8845	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℝ ↑_m 𝐼) → 𝐹:𝐼⟶ℝ)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐹:𝐼⟶ℝ)
13	12	ffvelcdmda 7080	. . . . . . 7 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
14	4, 5	eleqtrdi 2837	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐺 ∈ (ℝ ↑_m 𝐼))
15		elmapi 8845	. . . . . . . . 9 ⊢ (𝐺 ∈ (ℝ ↑_m 𝐼) → 𝐺:𝐼⟶ℝ)
16	14, 15	syl 17	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐺:𝐼⟶ℝ)
17	16	ffvelcdmda 7080	. . . . . . 7 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
18	13, 17	resubcld 11646	. . . . . 6 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
19	18	resqcld 14095	. . . . 5 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℝ)
20		simprl 768	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℝ⁺)
21	20	rpred 13022	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℝ)
22	21	resqcld 14095	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅↑2) ∈ ℝ)
23	22	adantr 480	. . . . 5 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑅↑2) ∈ ℝ)
24		absresq 15255	. . . . . . 7 ⊢ (((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ → ((abs‘((𝐹‘𝑘) − (𝐺‘𝑘)))↑2) = (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
25	18, 24	syl 17	. . . . . 6 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → ((abs‘((𝐹‘𝑘) − (𝐺‘𝑘)))↑2) = (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
26		rrndstprj1.1	. . . . . . . . . 10 ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))
27	26	remetdval 24660	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ (𝐺‘𝑘) ∈ ℝ) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) = (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))))
28	13, 17, 27	syl2anc 583	. . . . . . . 8 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) = (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))))
29		simprr 770	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)
30		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
32	30, 31	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) = ((𝐹‘𝑘)𝑀(𝐺‘𝑘)))
33	32	breq1d 5151	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅 ↔ ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅))
34	33	rspccva 3605	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅 ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅)
35	29, 34	sylan 579	. . . . . . . 8 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅)
36	28, 35	eqbrtrrd 5165	. . . . . . 7 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))) < 𝑅)
37	18	recnd 11246	. . . . . . . . 9 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℂ)
38	37	abscld 15389	. . . . . . . 8 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))) ∈ ℝ)
39	21	adantr 480	. . . . . . . 8 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ ℝ)
40	37	absge0d 15397	. . . . . . . 8 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → 0 ≤ (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))))
41	20	rpge0d 13026	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ 𝑅)
42	41	adantr 480	. . . . . . . 8 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → 0 ≤ 𝑅)
43	38, 39, 40, 42	lt2sqd 14224	. . . . . . 7 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → ((abs‘((𝐹‘𝑘) − (𝐺‘𝑘))) < 𝑅 ↔ ((abs‘((𝐹‘𝑘) − (𝐺‘𝑘)))↑2) < (𝑅↑2)))
44	36, 43	mpbid 231	. . . . . 6 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → ((abs‘((𝐹‘𝑘) − (𝐺‘𝑘)))↑2) < (𝑅↑2))
45	25, 44	eqbrtrrd 5165	. . . . 5 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) < (𝑅↑2))
46	2, 9, 19, 23, 45	fsumlt 15752	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) < Σ𝑘 ∈ 𝐼 (𝑅↑2))
47	2, 19	fsumrecl 15686	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℝ)
48	18	sqge0d 14107	. . . . . 6 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → 0 ≤ (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
49	2, 19, 48	fsumge0 15747	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
50		resqrtth 15208	. . . . 5 ⊢ ((Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) → ((√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))↑2) = Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
51	47, 49, 50	syl2anc 583	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))↑2) = Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
52		hashnncl 14331	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
53	2, 52	syl 17	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((♯‘𝐼) ∈ ℕ ↔ 𝐼 ≠ ∅))
54	9, 53	mpbird 257	. . . . . . . . . 10 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℕ)
55	54	nnrpd 13020	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℝ⁺)
56	55	rpred 13022	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℝ)
57	55	rpge0d 13026	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ (♯‘𝐼))
58		resqrtth 15208	. . . . . . . 8 ⊢ (((♯‘𝐼) ∈ ℝ ∧ 0 ≤ (♯‘𝐼)) → ((√‘(♯‘𝐼))↑2) = (♯‘𝐼))
59	56, 57, 58	syl2anc 583	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((√‘(♯‘𝐼))↑2) = (♯‘𝐼))
60	59	oveq2d 7421	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((𝑅↑2) · ((√‘(♯‘𝐼))↑2)) = ((𝑅↑2) · (♯‘𝐼)))
61	22	recnd 11246	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅↑2) ∈ ℂ)
62	55	rpcnd 13024	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℂ)
63	61, 62	mulcomd 11239	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((𝑅↑2) · (♯‘𝐼)) = ((♯‘𝐼) · (𝑅↑2)))
64	60, 63	eqtrd 2766	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((𝑅↑2) · ((√‘(♯‘𝐼))↑2)) = ((♯‘𝐼) · (𝑅↑2)))
65	20	rpcnd 13024	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℂ)
66	55	rpsqrtcld 15364	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
67	66	rpcnd 13024	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘(♯‘𝐼)) ∈ ℂ)
68	65, 67	sqmuld 14128	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((𝑅 · (√‘(♯‘𝐼)))↑2) = ((𝑅↑2) · ((√‘(♯‘𝐼))↑2)))
69		fsumconst 15742	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ (𝑅↑2) ∈ ℂ) → Σ𝑘 ∈ 𝐼 (𝑅↑2) = ((♯‘𝐼) · (𝑅↑2)))
70	2, 61, 69	syl2anc 583	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → Σ𝑘 ∈ 𝐼 (𝑅↑2) = ((♯‘𝐼) · (𝑅↑2)))
71	64, 68, 70	3eqtr4d 2776	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((𝑅 · (√‘(♯‘𝐼)))↑2) = Σ𝑘 ∈ 𝐼 (𝑅↑2))
72	46, 51, 71	3brtr4d 5173	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))↑2) < ((𝑅 · (√‘(♯‘𝐼)))↑2))
73	47, 49	resqrtcld 15370	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) ∈ ℝ)
74	20, 66	rpmulcld 13038	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅 · (√‘(♯‘𝐼))) ∈ ℝ⁺)
75	74	rpred 13022	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅 · (√‘(♯‘𝐼))) ∈ ℝ)
76	47, 49	sqrtge0d 15373	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
77	74	rpge0d 13026	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ (𝑅 · (√‘(♯‘𝐼))))
78	73, 75, 76, 77	lt2sqd 14224	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) < (𝑅 · (√‘(♯‘𝐼))) ↔ ((√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))↑2) < ((𝑅 · (√‘(♯‘𝐼)))↑2)))
79	72, 78	mpbird 257	. 2 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) < (𝑅 · (√‘(♯‘𝐼))))
80	7, 79	eqbrtrd 5163	1 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝐹(ℝⁿ‘𝐼)𝐺) < (𝑅 · (√‘(♯‘𝐼))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ∖ cdif 3940 ∅c0 4317 {csn 4623 class class class wbr 5141 × cxp 5667 ↾ cres 5671 ∘ ccom 5673 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ↑_m cmap 8822 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 ℕcn 12216 2c2 12271 ℝ⁺crp 12980 ↑cexp 14032 ♯chash 14295 √csqrt 15186 abscabs 15187 Σcsu 15638 ℝⁿcrrn 37206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-ico 13336 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-rrn 37207

This theorem is referenced by: rrncmslem 37213 rrnequiv 37216

W3C validator