rrndstprj2 - Mathbox for Jeff Madsen

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Theorem rrndstprj2 37542

Description: Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 37541 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 3958	. . 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 37539	. . 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 4789	. . . . . 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 2836	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐹 ∈ (ℝ ↑_m 𝐼))
11		elmapi 8867	. . . . . . . . 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 7087	. . . . . . 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 2836	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐺 ∈ (ℝ ↑_m 𝐼))
15		elmapi 8867	. . . . . . . . 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 7087	. . . . . . 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 11680	. . . . . 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 14135	. . . . 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 769	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℝ⁺)
21	20	rpred 13061	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℝ)
22	21	resqcld 14135	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅↑2) ∈ ℝ)
23	22	adantr 479	. . . . 5 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑅↑2) ∈ ℝ)
24		absresq 15299	. . . . . . 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 24790	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ (𝐺‘𝑘) ∈ ℝ) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) = (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))))
28	13, 17, 27	syl2anc 582	. . . . . . . 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 771	. . . . . . . . 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 6890	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
32	30, 31	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) = ((𝐹‘𝑘)𝑀(𝐺‘𝑘)))
33	32	breq1d 5153	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅 ↔ ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅))
34	33	rspccva 3606	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅 ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅)
35	29, 34	sylan 578	. . . . . . . 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 5167	. . . . . . 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 11280	. . . . . . . . 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 15433	. . . . . . . 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 479	. . . . . . . 8 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ ℝ)
40	37	absge0d 15441	. . . . . . . 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 13065	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ 𝑅)
42	41	adantr 479	. . . . . . . 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 14265	. . . . . . 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 5167	. . . . 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 15796	. . . 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 15730	. . . . 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 14147	. . . . . 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 15791	. . . . 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 15252	. . . . 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 582	. . . 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 14375	. . . . . . . . . . . 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 256	. . . . . . . . . 10 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℕ)
55	54	nnrpd 13059	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℝ⁺)
56	55	rpred 13061	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℝ)
57	55	rpge0d 13065	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ (♯‘𝐼))
58		resqrtth 15252	. . . . . . . 8 ⊢ (((♯‘𝐼) ∈ ℝ ∧ 0 ≤ (♯‘𝐼)) → ((√‘(♯‘𝐼))↑2) = (♯‘𝐼))
59	56, 57, 58	syl2anc 582	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((√‘(♯‘𝐼))↑2) = (♯‘𝐼))
60	59	oveq2d 7429	. . . . . 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 11280	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅↑2) ∈ ℂ)
62	55	rpcnd 13063	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℂ)
63	61, 62	mulcomd 11273	. . . . . 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 13063	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℂ)
66	55	rpsqrtcld 15408	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘(♯‘𝐼)) ∈ ℝ⁺)
67	66	rpcnd 13063	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘(♯‘𝐼)) ∈ ℂ)
68	65, 67	sqmuld 14168	. . . . 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 15786	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ (𝑅↑2) ∈ ℂ) → Σ𝑘 ∈ 𝐼 (𝑅↑2) = ((♯‘𝐼) · (𝑅↑2)))
70	2, 61, 69	syl2anc 582	. . . . 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 5175	. . 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 15414	. . . 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 13077	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅 · (√‘(♯‘𝐼))) ∈ ℝ⁺)
75	74	rpred 13061	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅 · (√‘(♯‘𝐼))) ∈ ℝ)
76	47, 49	sqrtge0d 15417	. . . 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 13065	. . . 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 14265	. . 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 256	. 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 5165	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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∖ cdif 3943 ∅c0 4322 {csn 4623 class class class wbr 5143 × cxp 5670 ↾ cres 5674 ∘ ccom 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 ↑_m cmap 8844 Fincfn 8963 ℂcc 11144 ℝcr 11145 0cc0 11146 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 ℕcn 12255 2c2 12310 ℝ⁺crp 13019 ↑cexp 14072 ♯chash 14339 √csqrt 15230 abscabs 15231 Σcsu 15682 ℝⁿcrrn 37536

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-inf2 9674 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-n0 12516 df-z 12602 df-uz 12866 df-rp 13020 df-ico 13375 df-fz 13530 df-fzo 13673 df-seq 14013 df-exp 14073 df-hash 14340 df-cj 15096 df-re 15097 df-im 15098 df-sqrt 15232 df-abs 15233 df-clim 15482 df-sum 15683 df-rrn 37537

This theorem is referenced by: rrncmslem 37543 rrnequiv 37546

W3C validator