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Theorem rrxmet 24772

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 769	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
3	1, 2	rrxfsupp 24766	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ∈ Fin)
4		simprr 771	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
5	1, 4	rrxfsupp 24766	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ∈ Fin)
6		unfi 9116	. . . . . . . 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 24767	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ⊆ 𝐼)
9	1, 4	rrxsuppss 24767	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ⊆ 𝐼)
10	8, 9	unssd 4146	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ 𝐼)
11	10	sselda 3944	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))) → 𝑘 ∈ 𝐼)
12	1, 2	rrxf 24765	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥:𝐼⟶ℝ)
13	12	ffvelcdmda 7035	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℝ)
14	1, 4	rrxf 24765	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦:𝐼⟶ℝ)
15	14	ffvelcdmda 7035	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℝ)
16	13, 15	resubcld 11583	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝑥‘𝑘) − (𝑦‘𝑘)) ∈ ℝ)
17	16	resqcld 14030	. . . . . . . 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 15619	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2) ∈ ℝ)
20	16	sqge0d 14042	. . . . . . . 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 15680	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))
23	19, 22	resqrtcld 15302	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ ℝ)
24	23	ralrimivva 3197	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ ℝ)
25		eqid 2736	. . . . 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 8000	. . . 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 217	. . 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 24770	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑥 supp 0) ∪ (𝑦 supp 0))(((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
30	29	feq1d 6653	. . 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 256	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐷:(𝑋 × 𝑋)⟶ℝ)
32		sqrt00 15148	. . . . . . 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 15683	. . . . . 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 11183	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝑥‘𝑘) − (𝑦‘𝑘)) ∈ ℂ)
36		sqeq0 14025	. . . . . . . . . 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 11183	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℂ)
39	15	recnd 11183	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℂ)
40	38, 39	subeq0ad 11522	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (((𝑥‘𝑘) − (𝑦‘𝑘)) = 0 ↔ (𝑥‘𝑘) = (𝑦‘𝑘)))
41	37, 40	bitrd 278	. . . . . . . 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 3172	. . . . . 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 304	. . . . 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 24769	. . . . . . 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 2738	. . . . 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 6669	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 Fn 𝐼)
49	14	ffnd 6669	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 Fn 𝐼)
50		eqfnfv 6982	. . . . . . 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 4132	. . . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐼 ∈ 𝑉)
55		0red 11158	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ∈ ℝ)
56	12, 53, 54, 55	suppssr 8127	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑥‘𝑘) = 0)
57		ssun2 4133	. . . . . . . . . . 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 8127	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑦‘𝑘) = 0)
60	56, 59	eqtr4d 2779	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))) → (𝑥‘𝑘) = (𝑦‘𝑘))
61	60	ralrimiva 3143	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑘 ∈ (𝐼 ∖ ((𝑥 supp 0) ∪ (𝑦 supp 0)))(𝑥‘𝑘) = (𝑦‘𝑘))
62	10, 61	raldifeq 4451	. . . . . 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 281	. . . . 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 310	. . . 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 24766	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 supp 0) ∈ Fin)
68		unfi 9116	. . . . . . . . . . 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 650	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ∈ Fin)
72	10	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ 𝐼)
73		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
74	1, 73	rrxsuppss 24767	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑧 supp 0) ⊆ 𝐼)
75	72, 74	unssd 4146	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ⊆ 𝐼)
76	75	sselda 3944	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0))) → 𝑘 ∈ 𝐼)
77	13	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℝ)
78	1, 73	rrxf 24765	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → 𝑧:𝐼⟶ℝ)
79	78	ffvelcdmda 7035	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑧‘𝑘) ∈ ℝ)
80	77, 79	resubcld 11583	. . . . . . . . 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 713	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℝ)
83	79, 82	resubcld 11583	. . . . . . . . 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 24764	. . . . . . 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 713	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥‘𝑘) ∈ ℂ)
87	79	recnd 11183	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑧‘𝑘) ∈ ℂ)
88	39	adantlr 713	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑦‘𝑘) ∈ ℂ)
89	86, 87, 88	npncand 11536	. . . . . . . . . . 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 7372	. . . . . . . . . 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 15588	. . . . . . . 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 6846	. . . . . . 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 14022	. . . . . . . . . . . 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 15588	. . . . . . . . 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 6846	. . . . . . . 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 7372	. . . . . . 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 5136	. . . . . 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 481	. . . . . . 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 24767	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥 supp 0) ⊆ 𝐼)
106	1, 104	rrxsuppss 24767	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑦 supp 0) ⊆ 𝐼)
107	105, 106	unssd 4146	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 supp 0) ∪ (𝑦 supp 0)) ⊆ 𝐼)
108	1, 66	rrxsuppss 24767	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 supp 0) ⊆ 𝐼)
109	107, 108	unssd 4146	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥 supp 0) ∪ (𝑦 supp 0)) ∪ (𝑧 supp 0)) ⊆ 𝐼)
110		ssun1 4132	. . . . . . . . . . . 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 24771	. . . . . . . . . 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 6846	. . . . . . . . 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 650	. . . . . . 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 2776	. . . . . 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 24769	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧𝐷𝑥) = (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑥 supp 0))(((𝑧‘𝑘) − (𝑥‘𝑘))↑2)))
118	117	3adant3r 1181	. . . . . . . . . 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 24769	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝐷𝑦) = (√‘Σ𝑘 ∈ ((𝑧 supp 0) ∪ (𝑦 supp 0))(((𝑧‘𝑘) − (𝑦‘𝑘))↑2)))
120	119	3adant3l 1180	. . . . . . . . . 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 7375	. . . . . . . . 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 4133	. . . . . . . . . . . . . 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 4146	. . . . . . . . . . . 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 24771	. . . . . . . . . . 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 6846	. . . . . . . . . 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 4146	. . . . . . . . . . . 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 24771	. . . . . . . . . . 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 6846	. . . . . . . . . 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 7375	. . . . . . . . 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 2776	. . . . . . . 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 650	. . . . . 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 5137	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
139	138	ralrimiva 3143	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))
140	64, 139	jca 512	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
141	140	ralrimivva 3197	. 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))
142		ovex 7390	. . . 4 ⊢ (ℝ ↑_m 𝐼) ∈ V
143	1, 142	rabex2 5291	. . 3 ⊢ 𝑋 ∈ V
144		ismet 23676	. . 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 {crab 3407 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 class class class wbr 5105 × cxp 5631 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 supp csupp 8092 ↑_m cmap 8765 Fincfn 8883 finSupp cfsupp 9305 ℂcc 11049 ℝcr 11050 0cc0 11051 + caddc 11054 ≤ cle 11190 − cmin 11385 2c2 12208 ↑cexp 13967 √csqrt 15118 Σcsu 15570 distcds 17142 Metcmet 20782 ℝ^crrx 24747

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-ico 13270 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-field 20188 df-subrg 20220 df-staf 20304 df-srng 20305 df-lmod 20324 df-lss 20393 df-sra 20633 df-rgmod 20634 df-met 20790 df-cnfld 20797 df-refld 21009 df-dsmm 21138 df-frlm 21153 df-nm 23938 df-tng 23940 df-tcph 24533 df-rrx 24749

This theorem is referenced by: rrxdstprj1 24773 rrxmetfi 24776

W3C validator