rrndstprj2 - Mathbox for Jeff Madsen

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Theorem rrndstprj2 38152

Description: Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 38151 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 1193	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐼 ∈ (Fin ∖ {∅}))
2	1	eldifad 3901	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐼 ∈ Fin)
3		simpl2 1194	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐹 ∈ 𝑋)
4		simpl3 1195	. . 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 38149	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(ℝⁿ‘𝐼)𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
7	2, 3, 4, 6	syl3anc 1374	. 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 4735	. . . . . 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 2846	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐹 ∈ (ℝ ↑_m 𝐼))
11		elmapi 8796	. . . . . . . . 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 7036	. . . . . . 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 2846	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐺 ∈ (ℝ ↑_m 𝐼))
15		elmapi 8796	. . . . . . . . 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 7036	. . . . . . 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 11578	. . . . . 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 14087	. . . . 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 771	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℝ⁺)
21	20	rpred 12986	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℝ)
22	21	resqcld 14087	. . . . . 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 15264	. . . . . . 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 24754	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ (𝐺‘𝑘) ∈ ℝ) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) = (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))))
28	13, 17, 27	syl2anc 585	. . . . . . . 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 773	. . . . . . . . 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 6840	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
32	30, 31	oveq12d 7385	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) = ((𝐹‘𝑘)𝑀(𝐺‘𝑘)))
33	32	breq1d 5095	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅 ↔ ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅))
34	33	rspccva 3563	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅 ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅)
35	29, 34	sylan 581	. . . . . . . 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 5109	. . . . . . 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 11173	. . . . . . . . 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 15401	. . . . . . . 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 15409	. . . . . . . 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 12990	. . . . . . . . 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 14218	. . . . . . 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 232	. . . . . 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 5109	. . . . 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 15763	. . . 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 15696	. . . . 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 14099	. . . . . 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 15758	. . . . 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 15217	. . . . 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 585	. . . 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 14328	. . . . . . . . . . . 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 12984	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℝ⁺)
56	55	rpred 12986	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℝ)
57	55	rpge0d 12990	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ (♯‘𝐼))
58		resqrtth 15217	. . . . . . . 8 ⊢ (((♯‘𝐼) ∈ ℝ ∧ 0 ≤ (♯‘𝐼)) → ((√‘(♯‘𝐼))↑2) = (♯‘𝐼))
59	56, 57, 58	syl2anc 585	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((√‘(♯‘𝐼))↑2) = (♯‘𝐼))
60	59	oveq2d 7383	. . . . . 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 11173	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅↑2) ∈ ℂ)
62	55	rpcnd 12988	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℂ)
63	61, 62	mulcomd 11166	. . . . . 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 2771	. . . . 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 12988	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℂ)
66	55	rpsqrtcld 15374	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
67	66	rpcnd 12988	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘(♯‘𝐼)) ∈ ℂ)
68	65, 67	sqmuld 14120	. . . . 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 15752	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ (𝑅↑2) ∈ ℂ) → Σ𝑘 ∈ 𝐼 (𝑅↑2) = ((♯‘𝐼) · (𝑅↑2)))
70	2, 61, 69	syl2anc 585	. . . . 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 2781	. . . 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 5117	. . 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 15380	. . . 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 13002	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅 · (√‘(♯‘𝐼))) ∈ ℝ⁺)
75	74	rpred 12986	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅 · (√‘(♯‘𝐼))) ∈ ℝ)
76	47, 49	sqrtge0d 15383	. . . 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 12990	. . . 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 14218	. . 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 5107	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∖ cdif 3886 ∅c0 4273 {csn 4567 class class class wbr 5085 × cxp 5629 ↾ cres 5633 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 Fincfn 8893 ℂcc 11036 ℝcr 11037 0cc0 11038 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 ℕcn 12174 2c2 12236 ℝ⁺crp 12942 ↑cexp 14023 ♯chash 14292 √csqrt 15195 abscabs 15196 Σcsu 15648 ℝⁿcrrn 38146

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-ico 13304 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-rrn 38147

This theorem is referenced by: rrncmslem 38153 rrnequiv 38156

W3C validator