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Theorem rrxmet 22999

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 790	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
3	1, 2	rrxfsupp 22993	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ∈ Fin)
4		simprr 792	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
5	1, 4	rrxfsupp 22993	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ∈ Fin)
6		unfi 8112	. . . . . . . 8 ⊢ (((𝑥 supp 0) ∈ Fin ∧ (𝑦 supp 0) ∈ Fin) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ∈ Fin)
7	3, 5, 6	syl2anc 691	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ∈ Fin)
8	1, 2	rrxsuppss 22994	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ⊆ 𝐼)
9	1, 4	rrxsuppss 22994	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ⊆ 𝐼)
10	8, 9	unssd 3751	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ 𝐼)
11	10	sselda 3568	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))) → 𝑘 ∈ 𝐼)
12	1, 2	rrxf 22992	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥:𝐼⟶ℝ)
13	12	ffvelrnda 6267	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℝ)
14	1, 4	rrxf 22992	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦:𝐼⟶ℝ)
15	14	ffvelrnda 6267	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℝ)
16	13, 15	resubcld 10337	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝑥‘𝑘) − (𝑦‘𝑘)) ∈ ℝ)
17	16	resqcld 12897	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) ∈ ℝ)
18	11, 17	syldan 486	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))) → (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) ∈ ℝ)
19	7, 18	fsumrecl 14312	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) ∈ ℝ)
20	16	sqge0d 12898	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → 0 ≤ (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
21	11, 20	syldan 486	. . . . . . 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 14368	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
23	19, 22	resqrtcld 14004	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ ℝ)
24	23	ralrimivva 2954	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ ℝ)
25		eqid 2610	. . . . 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	fmpt2 7126	. . . 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 207	. . 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 22997	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
30	29	feq1d 5943	. . 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 246	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐷:(𝑋 × 𝑋)⟶ℝ)
32		sqrt00 13852	. . . . . . 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 691	. . . . . 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 14371	. . . . . 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 9947	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝑥‘𝑘) − (𝑦‘𝑘)) ∈ ℂ)
36		sqeq0 12789	. . . . . . . . . 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 9947	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℂ)
39	15	recnd 9947	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℂ)
40	38, 39	subeq0ad 10281	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (((𝑥‘𝑘) − (𝑦‘𝑘)) = 0 ↔ (𝑥‘𝑘) = (𝑦‘𝑘)))
41	37, 40	bitrd 267	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = 0 ↔ (𝑥‘𝑘) = (𝑦‘𝑘)))
42	11, 41	syldan 486	. . . . . . 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 2968	. . . . . 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 293	. . . . 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 22996	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
46	45	3expb 1258	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))
47	46	eqeq1d 2612	. . . . 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 5959	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 Fn 𝐼)
49	14	ffnd 5959	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 Fn 𝐼)
50		eqfnfv 6219	. . . . . . 7 ⊢ ((𝑥 Fn 𝐼 ∧ 𝑦 Fn 𝐼) → (𝑥 = 𝑦 ↔ ∀𝑘 ∈ 𝐼 (𝑥‘𝑘) = (𝑦‘𝑘)))
51	48, 49, 50	syl2anc 691	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 = 𝑦 ↔ ∀𝑘 ∈ 𝐼 (𝑥‘𝑘) = (𝑦‘𝑘)))
52		ssun1 3738	. . . . . . . . . . 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 472	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐼 ∈ 𝑉)
55		0red 9920	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ∈ ℝ)
56	12, 53, 54, 55	suppssr 7213	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑥‘𝑘) = 0)
57		ssun2 3739	. . . . . . . . . . 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 7213	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑦‘𝑘) = 0)
60	56, 59	eqtr4d 2647	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑥‘𝑘) = (𝑦‘𝑘))
61	60	ralrimiva 2949	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))(𝑥‘𝑘) = (𝑦‘𝑘))
62	10, 61	raldifeq 4011	. . . . . 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 270	. . . . 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 299	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
65	7	3adant2 1073	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ∈ Fin)
66		simp2 1055	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
67	1, 66	rrxfsupp 22993	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 supp 0) ∈ Fin)
68		unfi 8112	. . . . . . . . . . 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 691	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ∈ Fin)
70	69	3expa 1257	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ∈ Fin)
71	70	an32s 842	. . . . . . . 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 476	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
74	1, 73	rrxsuppss 22994	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑧 supp 0) ⊆ 𝐼)
75	72, 74	unssd 3751	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ⊆ 𝐼)
76	75	sselda 3568	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))) → 𝑘 ∈ 𝐼)
77	13	adantlr 747	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℝ)
78	1, 73	rrxf 22992	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → 𝑧:𝐼⟶ℝ)
79	78	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑧‘𝑘) ∈ ℝ)
80	77, 79	resubcld 10337	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝑥‘𝑘) − (𝑧‘𝑘)) ∈ ℝ)
81	76, 80	syldan 486	. . . . . . . 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 747	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℝ)
83	79, 82	resubcld 10337	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝑧‘𝑘) − (𝑦‘𝑘)) ∈ ℝ)
84	76, 83	syldan 486	. . . . . . . 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 22991	. . . . . . 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 747	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℂ)
87	79	recnd 9947	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑧‘𝑘) ∈ ℂ)
88	39	adantlr 747	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℂ)
89	86, 87, 88	npncand 10295	. . . . . . . . . . 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 6564	. . . . . . . . . 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 486	. . . . . . . . 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 14281	. . . . . . . 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 6107	. . . . . . 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 12786	. . . . . . . . . . . 12 ⊢ (((𝑥‘𝑘) ∈ ℂ ∧ (𝑧‘𝑘) ∈ ℂ) → (((𝑥‘𝑘) − (𝑧‘𝑘))↑2) = (((𝑧‘𝑘) − (𝑥‘𝑘))↑2))
95	86, 87, 94	syl2anc 691	. . . . . . . . . . 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 486	. . . . . . . . . 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 14281	. . . . . . . . 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 6107	. . . . . . . 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 6564	. . . . . . 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 4614	. . . . . 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 1054	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐼 ∈ 𝑉)
103	2	3adant2 1073	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
104	4	3adant2 1073	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
105	1, 103	rrxsuppss 22994	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ⊆ 𝐼)
106	1, 104	rrxsuppss 22994	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ⊆ 𝐼)
107	105, 106	unssd 3751	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ 𝐼)
108	1, 66	rrxsuppss 22994	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 supp 0) ⊆ 𝐼)
109	107, 108	unssd 3751	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ⊆ 𝐼)
110		ssun1 3738	. . . . . . . . . . . 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 22998	. . . . . . . . . 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 6107	. . . . . . . . 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 1257	. . . . . . . 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 842	. . . . . . 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 2644	. . . . . 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 22996	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑥 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)))
118	117	3adant3r 1315	. . . . . . . . . 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 22996	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑦 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2)))
120	119	3adant3l 1314	. . . . . . . . . 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 6567	. . . . . . . . 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 3739	. . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . 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 3751	. . . . . . . . . . . 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 22998	. . . . . . . . . . 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 6107	. . . . . . . . . 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 3577	. . . . . . . . . . . . . 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 3751	. . . . . . . . . . . 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 22998	. . . . . . . . . . 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 6107	. . . . . . . . . 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 6567	. . . . . . . . 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 2644	. . . . . . . 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 1257	. . . . . . 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 842	. . . . . 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 4615	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
139	138	ralrimiva 2949	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
140	64, 139	jca 553	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
141	140	ralrimivva 2954	. 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
142		ovex 6577	. . . 4 ⊢ (ℝ ↑_𝑚 𝐼) ∈ V
143	1, 142	rabex2 4742	. . 3 ⊢ 𝑋 ∈ V
144		ismet 21938	. . 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 695	1 ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 supp csupp 7182 ↑_𝑚 cmap 7744 Fincfn 7841 finSupp cfsupp 8158 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 ≤ cle 9954 − cmin 10145 2c2 10947 ↑cexp 12722 √csqrt 13821 Σcsu 14264 distcds 15777 Metcme 19553 ℝ^crrx 22979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-field 18573 df-subrg 18601 df-staf 18668 df-srng 18669 df-lmod 18688 df-lss 18754 df-sra 18993 df-rgmod 18994 df-met 19561 df-cnfld 19568 df-refld 19770 df-dsmm 19895 df-frlm 19910 df-nm 22197 df-tng 22199 df-tch 22777 df-rrx 22981

This theorem is referenced by: rrxdstprj1 23000 rrxmetfi 39183

W3C validator