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Theorem rrxmet 25341

Description: Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.)

**Hypotheses**
Ref	Expression
rrxmval.1	⊢ 𝑋 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
rrxmval.d	⊢ 𝐷 = (dist‘(ℝ^‘𝐼))

**Assertion**
Ref	Expression
rrxmet	⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

Distinct variable groups: ℎ,𝐼 ℎ,𝑉 Allowed substitution hints: 𝐷(ℎ) 𝑋(ℎ)

Proof of Theorem rrxmet Dummy variables 𝑘 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxmval.1	. . . . . . . . 9 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
2		simprl 770	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
3	1, 2	rrxfsupp 25335	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ∈ Fin)
4		simprr 772	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
5	1, 4	rrxfsupp 25335	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ∈ Fin)
6		unfi 9112	. . . . . . . 8 ⊢ (((𝑥 supp 0) ∈ Fin ∧ (𝑦 supp 0) ∈ Fin) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ∈ Fin)
7	3, 5, 6	syl2anc 584	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ∈ Fin)
8	1, 2	rrxsuppss 25336	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ⊆ 𝐼)
9	1, 4	rrxsuppss 25336	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ⊆ 𝐼)
10	8, 9	unssd 4151	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ 𝐼)
11	10	sselda 3943	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))) → 𝑘 ∈ 𝐼)
12	1, 2	rrxf 25334	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥:𝐼⟶ℝ)
13	12	ffvelcdmda 7038	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℝ)
14	1, 4	rrxf 25334	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦:𝐼⟶ℝ)
15	14	ffvelcdmda 7038	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℝ)
16	13, 15	resubcld 11582	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝑥‘𝑘) − (𝑦‘𝑘)) ∈ ℝ)
17	16	resqcld 14066	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) ∈ ℝ)
18	11, 17	syldan 591	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))) → (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) ∈ ℝ)
19	7, 18	fsumrecl 15676	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) ∈ ℝ)
20	16	sqge0d 14078	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → 0 ≤ (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
21	11, 20	syldan 591	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))) → 0 ≤ (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
22	7, 18, 21	fsumge0 15737	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
23	19, 22	resqrtcld 15360	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ ℝ)
24	23	ralrimivva 3178	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ ℝ)
25		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
26	25	fmpo 8026	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ ℝ ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))):(𝑋 × 𝑋)⟶ℝ)
27	24, 26	sylib 218	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))):(𝑋 × 𝑋)⟶ℝ)
28		rrxmval.d	. . . . 5 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
29	1, 28	rrxmfval 25339	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
30	29	feq1d 6652	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))):(𝑋 × 𝑋)⟶ℝ))
31	27, 30	mpbird 257	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐷:(𝑋 × 𝑋)⟶ℝ)
32		sqrt00 15205	. . . . . . 7 ⊢ ((Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) → ((√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = 0 ↔ Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0))
33	19, 22, 32	syl2anc 584	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = 0 ↔ Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0))
34	7, 18, 21	fsum00 15740	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0 ↔ ∀𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0))
35	16	recnd 11178	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝑥‘𝑘) − (𝑦‘𝑘)) ∈ ℂ)
36		sqeq0 14061	. . . . . . . . . 10 ⊢ (((𝑥‘𝑘) − (𝑦‘𝑘)) ∈ ℂ → ((((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0 ↔ ((𝑥‘𝑘) − (𝑦‘𝑘)) = 0))
37	35, 36	syl 17	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0 ↔ ((𝑥‘𝑘) − (𝑦‘𝑘)) = 0))
38	13	recnd 11178	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℂ)
39	15	recnd 11178	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℂ)
40	38, 39	subeq0ad 11519	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (((𝑥‘𝑘) − (𝑦‘𝑘)) = 0 ↔ (𝑥‘𝑘) = (𝑦‘𝑘)))
41	37, 40	bitrd 279	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0 ↔ (𝑥‘𝑘) = (𝑦‘𝑘)))
42	11, 41	syldan 591	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))) → ((((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0 ↔ (𝑥‘𝑘) = (𝑦‘𝑘)))
43	42	ralbidva 3154	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (∀𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0 ↔ ∀𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(𝑥‘𝑘) = (𝑦‘𝑘)))
44	33, 34, 43	3bitrd 305	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = 0 ↔ ∀𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(𝑥‘𝑘) = (𝑦‘𝑘)))
45	1, 28	rrxmval 25338	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
46	45	3expb 1120	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
47	46	eqeq1d 2731	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = 0))
48	12	ffnd 6671	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 Fn 𝐼)
49	14	ffnd 6671	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 Fn 𝐼)
50		eqfnfv 6985	. . . . . . 7 ⊢ ((𝑥 Fn 𝐼 ∧ 𝑦 Fn 𝐼) → (𝑥 = 𝑦 ↔ ∀𝑘 ∈ 𝐼 (𝑥‘𝑘) = (𝑦‘𝑘)))
51	48, 49, 50	syl2anc 584	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 = 𝑦 ↔ ∀𝑘 ∈ 𝐼 (𝑥‘𝑘) = (𝑦‘𝑘)))
52		ssun1 4137	. . . . . . . . . . 11 ⊢ (𝑥 supp 0) ⊆ ((𝑥 supp 0) ∪ (𝑦 supp 0))
53	52	a1i 11	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ⊆ ((𝑥 supp 0) ∪ (𝑦 supp 0)))
54		simpl 482	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐼 ∈ 𝑉)
55		0red 11153	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ∈ ℝ)
56	12, 53, 54, 55	suppssr 8151	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑥‘𝑘) = 0)
57		ssun2 4138	. . . . . . . . . . 11 ⊢ (𝑦 supp 0) ⊆ ((𝑥 supp 0) ∪ (𝑦 supp 0))
58	57	a1i 11	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ⊆ ((𝑥 supp 0) ∪ (𝑦 supp 0)))
59	14, 58, 54, 55	suppssr 8151	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑦‘𝑘) = 0)
60	56, 59	eqtr4d 2767	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑥‘𝑘) = (𝑦‘𝑘))
61	60	ralrimiva 3125	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))(𝑥‘𝑘) = (𝑦‘𝑘))
62	10, 61	raldifeq 4453	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (∀𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(𝑥‘𝑘) = (𝑦‘𝑘) ↔ ∀𝑘 ∈ 𝐼 (𝑥‘𝑘) = (𝑦‘𝑘)))
63	51, 62	bitr4d 282	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 = 𝑦 ↔ ∀𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(𝑥‘𝑘) = (𝑦‘𝑘)))
64	44, 47, 63	3bitr4d 311	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
65	7	3adant2 1131	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ∈ Fin)
66		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
67	1, 66	rrxfsupp 25335	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 supp 0) ∈ Fin)
68		unfi 9112	. . . . . . . . . . 11 ⊢ ((((𝑥 supp 0) ∪ (𝑦 supp 0)) ∈ Fin ∧ (𝑧 supp 0) ∈ Fin) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ∈ Fin)
69	65, 67, 68	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ∈ Fin)
70	69	3expa 1118	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ∈ Fin)
71	70	an32s 652	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ∈ Fin)
72	10	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ 𝐼)
73		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
74	1, 73	rrxsuppss 25336	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑧 supp 0) ⊆ 𝐼)
75	72, 74	unssd 4151	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ⊆ 𝐼)
76	75	sselda 3943	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))) → 𝑘 ∈ 𝐼)
77	13	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℝ)
78	1, 73	rrxf 25334	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → 𝑧:𝐼⟶ℝ)
79	78	ffvelcdmda 7038	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑧‘𝑘) ∈ ℝ)
80	77, 79	resubcld 11582	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝑥‘𝑘) − (𝑧‘𝑘)) ∈ ℝ)
81	76, 80	syldan 591	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))) → ((𝑥‘𝑘) − (𝑧‘𝑘)) ∈ ℝ)
82	15	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℝ)
83	79, 82	resubcld 11582	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝑧‘𝑘) − (𝑦‘𝑘)) ∈ ℝ)
84	76, 83	syldan 591	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))) → ((𝑧‘𝑘) − (𝑦‘𝑘)) ∈ ℝ)
85	71, 81, 84	trirn 25333	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))((((𝑥‘𝑘) − (𝑧‘𝑘)) + ((𝑧‘𝑘) − (𝑦‘𝑘)))↑2)) ≤ ((√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑧‘𝑘))↑2)) + (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))))
86	38	adantlr 715	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℂ)
87	79	recnd 11178	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑧‘𝑘) ∈ ℂ)
88	39	adantlr 715	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℂ)
89	86, 87, 88	npncand 11533	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝑥‘𝑘) − (𝑧‘𝑘)) + ((𝑧‘𝑘) − (𝑦‘𝑘))) = ((𝑥‘𝑘) − (𝑦‘𝑘)))
90	89	oveq1d 7384	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((((𝑥‘𝑘) − (𝑧‘𝑘)) + ((𝑧‘𝑘) − (𝑦‘𝑘)))↑2) = (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
91	76, 90	syldan 591	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))) → ((((𝑥‘𝑘) − (𝑧‘𝑘)) + ((𝑧‘𝑘) − (𝑦‘𝑘)))↑2) = (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
92	91	sumeq2dv 15644	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))((((𝑥‘𝑘) − (𝑧‘𝑘)) + ((𝑧‘𝑘) − (𝑦‘𝑘)))↑2) = Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
93	92	fveq2d 6844	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))((((𝑥‘𝑘) − (𝑧‘𝑘)) + ((𝑧‘𝑘) − (𝑦‘𝑘)))↑2)) = (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
94		sqsubswap 14058	. . . . . . . . . . . 12 ⊢ (((𝑥‘𝑘) ∈ ℂ ∧ (𝑧‘𝑘) ∈ ℂ) → (((𝑥‘𝑘) − (𝑧‘𝑘))↑2) = (((𝑧‘𝑘) − (𝑥‘𝑘))↑2))
95	86, 87, 94	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝑥‘𝑘) − (𝑧‘𝑘))↑2) = (((𝑧‘𝑘) − (𝑥‘𝑘))↑2))
96	76, 95	syldan 591	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))) → (((𝑥‘𝑘) − (𝑧‘𝑘))↑2) = (((𝑧‘𝑘) − (𝑥‘𝑘))↑2))
97	96	sumeq2dv 15644	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑧‘𝑘))↑2) = Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2))
98	97	fveq2d 6844	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑧‘𝑘))↑2)) = (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)))
99	98	oveq1d 7384	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → ((√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑧‘𝑘))↑2)) + (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))) = ((√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) + (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))))
100	85, 93, 99	3brtr3d 5133	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ≤ ((√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) + (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))))
101	46	adantr 480	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
102		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐼 ∈ 𝑉)
103	2	3adant2 1131	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
104	4	3adant2 1131	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
105	1, 103	rrxsuppss 25336	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ⊆ 𝐼)
106	1, 104	rrxsuppss 25336	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ⊆ 𝐼)
107	105, 106	unssd 4151	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ 𝐼)
108	1, 66	rrxsuppss 25336	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 supp 0) ⊆ 𝐼)
109	107, 108	unssd 4151	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ⊆ 𝐼)
110		ssun1 4137	. . . . . . . . . . . 12 ⊢ ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))
111	110	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)))
112	1, 28, 102, 103, 104, 109, 69, 111	rrxmetlem 25340	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
113	112	fveq2d 6844	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
114	113	3expa 1118	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
115	114	an32s 652	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
116	101, 115	eqtrd 2764	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) = (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
117	1, 28	rrxmval 25338	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑥 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)))
118	117	3adant3r 1182	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧𝐷𝑥) = (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑥 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)))
119	1, 28	rrxmval 25338	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑦 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2)))
120	119	3adant3l 1181	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑦 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2)))
121	118, 120	oveq12d 7387	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑥 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) + (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑦 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))))
122		ssun2 4138	. . . . . . . . . . . . . 14 ⊢ (𝑧 supp 0) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))
123	122	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 supp 0) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)))
124	52, 110	sstri 3953	. . . . . . . . . . . . . 14 ⊢ (𝑥 supp 0) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))
125	124	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)))
126	123, 125	unssd 4151	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑧 supp 0) ∪ (𝑥 supp 0)) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)))
127	1, 28, 102, 66, 103, 109, 69, 126	rrxmetlem 25340	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑥 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2) = Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2))
128	127	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑥 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) = (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)))
129	57, 110	sstri 3953	. . . . . . . . . . . . . 14 ⊢ (𝑦 supp 0) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))
130	129	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)))
131	123, 130	unssd 4151	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑧 supp 0) ∪ (𝑦 supp 0)) ⊆ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)))
132	1, 28, 102, 66, 104, 109, 69, 131	rrxmetlem 25340	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑦 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2) = Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))
133	132	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑦 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2)) = (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2)))
134	128, 133	oveq12d 7387	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑥 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) + (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑦 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))) = ((√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) + (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))))
135	121, 134	eqtrd 2764	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) + (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))))
136	135	3expa 1118	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) + (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))))
137	136	an32s 652	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)) = ((√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)) + (√‘Σ𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2))))
138	100, 116, 137	3brtr4d 5134	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
139	138	ralrimiva 3125	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
140	64, 139	jca 511	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
141	140	ralrimivva 3178	. 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
142		ovex 7402	. . . 4 ⊢ (ℝ ↑_m 𝐼) ∈ V
143	1, 142	rabex2 5291	. . 3 ⊢ 𝑋 ∈ V
144		ismet 24244	. . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
145	143, 144	ax-mp 5	. 2 ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
146	31, 141, 145	sylanbrc 583	1 ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 {crab 3402 Vcvv 3444 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 class class class wbr 5102 × cxp 5629 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 supp csupp 8116 ↑_m cmap 8776 Fincfn 8895 finSupp cfsupp 9288 ℂcc 11042 ℝcr 11043 0cc0 11044 + caddc 11047 ≤ cle 11185 − cmin 11381 2c2 12217 ↑cexp 14002 √csqrt 15175 Σcsu 15628 distcds 17205 Metcmet 21282 ℝ^crrx 25316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-rp 12928 df-ico 13288 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17380 df-gsum 17381 df-prds 17386 df-pws 17388 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-grp 18850 df-minusg 18851 df-sbg 18852 df-subg 19037 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-rhm 20392 df-subrng 20466 df-subrg 20490 df-drng 20651 df-field 20652 df-staf 20759 df-srng 20760 df-lmod 20800 df-lss 20870 df-sra 21112 df-rgmod 21113 df-met 21290 df-cnfld 21297 df-refld 21547 df-dsmm 21674 df-frlm 21689 df-nm 24503 df-tng 24505 df-tcph 25102 df-rrx 25318

This theorem is referenced by: rrxdstprj1 25342 rrxmetfi 25345

W3C validator