Theorem rrndstprj2 34249

Description: Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 34248 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 1199	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐼 ∈ (Fin ∖ {∅}))
2	1	eldifad 3803	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐼 ∈ Fin)
3		simpl2 1201	. . 3 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐹 ∈ 𝑋)
4		simpl3 1203	. . 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 34246	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(ℝn‘𝐼)𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
7	2, 3, 4, 6	syl3anc 1439	. 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 4553	. . . . . 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	syl6eleq 2868	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐹 ∈ (ℝ ↑𝑚 𝐼))
11		elmapi 8162	. . . . . . . . 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	ffvelrnda 6623	. . . . . . 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	syl6eleq 2868	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝐺 ∈ (ℝ ↑𝑚 𝐼))
15		elmapi 8162	. . . . . . . . 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	ffvelrnda 6623	. . . . . . 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 10803	. . . . . 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 13356	. . . . 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 761	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℝ+)
21	20	rpred 12181	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℝ)
22	21	resqcld 13356	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅↑2) ∈ ℝ)
23	22	adantr 474	. . . . 5 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → (𝑅↑2) ∈ ℝ)
24		absresq 14449	. . . . . . 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 23000	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ (𝐺‘𝑘) ∈ ℝ) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) = (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))))
28	13, 17, 27	syl2anc 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)) = (abs‘((𝐹‘𝑘) − (𝐺‘𝑘))))
29		simprr 763	. . . . . . . . 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 6446	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31		fveq2 6446	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
32	30, 31	oveq12d 6940	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) = ((𝐹‘𝑘)𝑀(𝐺‘𝑘)))
33	32	breq1d 4896	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅 ↔ ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅))
34	33	rspccva 3509	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅 ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘)𝑀(𝐺‘𝑘)) < 𝑅)
35	29, 34	sylan 575	. . . . . . . 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 4910	. . . . . . 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 10405	. . . . . . . . 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 14583	. . . . . . . 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 474	. . . . . . . 8 ⊢ ((((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ ℝ)
40	37	absge0d 14591	. . . . . . . 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 12185	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ 𝑅)
42	41	adantr 474	. . . . . . . 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 13364	. . . . . . 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 224	. . . . . 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 4910	. . . . 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 14936	. . . 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 14872	. . . . 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 13357	. . . . . 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 14931	. . . . 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 14403	. . . . 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 579	. . . 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 13472	. . . . . . . . . . . 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 249	. . . . . . . . . 10 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℕ)
55	54	nnrpd 12179	. . . . . . . . 9 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℝ+)
56	55	rpred 12181	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℝ)
57	55	rpge0d 12185	. . . . . . . 8 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 0 ≤ (♯‘𝐼))
58		resqrtth 14403	. . . . . . . 8 ⊢ (((♯‘𝐼) ∈ ℝ ∧ 0 ≤ (♯‘𝐼)) → ((√‘(♯‘𝐼))↑2) = (♯‘𝐼))
59	56, 57, 58	syl2anc 579	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → ((√‘(♯‘𝐼))↑2) = (♯‘𝐼))
60	59	oveq2d 6938	. . . . . 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 10405	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅↑2) ∈ ℂ)
62	55	rpcnd 12183	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (♯‘𝐼) ∈ ℂ)
63	61, 62	mulcomd 10398	. . . . . 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 2813	. . . . 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 12183	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → 𝑅 ∈ ℂ)
66	55	rpsqrtcld 14558	. . . . . . 7 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘(♯‘𝐼)) ∈ ℝ+)
67	66	rpcnd 12183	. . . . . 6 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (√‘(♯‘𝐼)) ∈ ℂ)
68	65, 67	sqmuld 13339	. . . . 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 14926	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ (𝑅↑2) ∈ ℂ) → Σ𝑘 ∈ 𝐼 (𝑅↑2) = ((♯‘𝐼) · (𝑅↑2)))
70	2, 61, 69	syl2anc 579	. . . . 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 2823	. . . 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 4918	. . 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 14564	. . . 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 12197	. . . . 5 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅 · (√‘(♯‘𝐼))) ∈ ℝ+)
75	74	rpred 12181	. . . 4 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝑅 · (√‘(♯‘𝐼))) ∈ ℝ)
76	47, 49	sqrtge0d 14567	. . . 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 12185	. . . 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 13364	. . 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 249	. 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 4908	1 ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝐹(ℝn‘𝐼)𝐺) < (𝑅 · (√‘(♯‘𝐼))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  ∀wral 3089   ∖ cdif 3788  ∅c0 4140  {csn 4397   class class class wbr 4886   × cxp 5353   ↾ cres 5357   ∘ ccom 5359  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ↑_𝑚 cmap 8140  Fincfn 8241  ℂcc 10270  ℝcr 10271  0cc0 10272   · cmul 10277   < clt 10411   ≤ cle 10412   − cmin 10606  ℕcn 11374  2c2 11430  ℝ⁺crp 12137  ↑cexp 13178  ♯chash 13435  √csqrt 14380  abscabs 14381  Σcsu 14824  ℝⁿcrrn 34243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-ico 12493  df-fz 12644  df-fzo 12785  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-rrn 34244

This theorem is referenced by:  rrncmslem  34250  rrnequiv  34253