Theorem brcgr 28983

Description: The binary relation form of the congruence predicate. The statement ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ should be read informally as "the 𝑁 dimensional point 𝐴 is as far from 𝐵 as 𝐶 is from 𝐷, or "the line segment 𝐴𝐵 is congruent to the line segment 𝐶𝐷. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)

Assertion

Ref	Expression
brcgr	⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))

Distinct variable groups:   𝑖,𝑁   𝐴,𝑖   𝐵,𝑖   𝐶,𝑖   𝐷,𝑖

Proof of Theorem brcgr

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5411	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5411	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eleq1 2825	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
4	3	anbi1d 632	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ↔ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))))
5		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
6	5	fveq1d 6836	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)‘𝑖) = ((1st ‘⟨𝐴, 𝐵⟩)‘𝑖))
7		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
8	7	fveq1d 6836	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((2nd ‘𝑥)‘𝑖) = ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))
9	6, 8	oveq12d 7378	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖)) = (((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖)))
10	9	oveq1d 7375	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = ((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2))
11	10	sumeq2sdv 15656	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2))
12	11	eqeq1d 2739	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2) ↔ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2)))
13	4, 12	anbi12d 633	. . . 4 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2)) ↔ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2))))
14	13	rexbidv 3162	. . 3 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2)) ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2))))
15		eleq1 2825	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
16	15	anbi2d 631	. . . . 5 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ↔ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))))
17		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
18	17	fveq1d 6836	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((1st ‘𝑦)‘𝑖) = ((1st ‘⟨𝐶, 𝐷⟩)‘𝑖))
19		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
20	19	fveq1d 6836	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((2nd ‘𝑦)‘𝑖) = ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))
21	18, 20	oveq12d 7378	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖)) = (((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖)))
22	21	oveq1d 7375	. . . . . . 7 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2) = ((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))
23	22	sumeq2sdv 15656	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))
24	23	eqeq2d 2748	. . . . 5 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2) ↔ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)))
25	16, 24	anbi12d 633	. . . 4 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2)) ↔ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))))
26	25	rexbidv 3162	. . 3 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2)) ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))))
27		df-cgr 28975	. . 3 ⊢ Cgr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2))}
28	1, 2, 14, 26, 27	brab 5491	. 2 ⊢ (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)))
29		opelxp2 5667	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → 𝐷 ∈ (𝔼‘𝑛))
30	29	ad2antll 730	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))) → 𝐷 ∈ (𝔼‘𝑛))
31		simplrr 778	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))) → 𝐷 ∈ (𝔼‘𝑁))
32		eedimeq 28981	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
33	30, 31, 32	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))) → 𝑛 = 𝑁)
34	33	adantlr 716	. . . . . . . 8 ⊢ (((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))) → 𝑛 = 𝑁)
35		oveq2 7368	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
36	35	sumeq1d 15653	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2))
37	35	sumeq1d 15653	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))
38	36, 37	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) ↔ Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)))
39	34, 38	syl 17	. . . . . . 7 ⊢ (((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))) → (Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) ↔ Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)))
40		op1stg 7947	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
41	40	fveq1d 6836	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) = (𝐴‘𝑖))
42		op2ndg 7948	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
43	42	fveq1d 6836	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖) = (𝐵‘𝑖))
44	41, 43	oveq12d 7378	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖)) = ((𝐴‘𝑖) − (𝐵‘𝑖)))
45	44	oveq1d 7375	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = (((𝐴‘𝑖) − (𝐵‘𝑖))↑2))
46	45	sumeq2sdv 15656	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2))
47		op1stg 7947	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
48	47	fveq1d 6836	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) = (𝐶‘𝑖))
49		op2ndg 7948	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
50	49	fveq1d 6836	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖) = (𝐷‘𝑖))
51	48, 50	oveq12d 7378	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖)) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
52	51	oveq1d 7375	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) = (((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
53	52	sumeq2sdv 15656	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
54	46, 53	eqeqan12d 2751	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
55	54	ad2antrr 727	. . . . . . 7 ⊢ (((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))) → (Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
56	39, 55	bitrd 279	. . . . . 6 ⊢ (((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))) → (Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
57	56	biimpd 229	. . . . 5 ⊢ (((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))) → (Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) → Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
58	57	expimpd 453	. . . 4 ⊢ ((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) → (((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)) → Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
59	58	rexlimdva 3139	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)) → Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
60		eleenn 28979	. . . . 5 ⊢ (𝐷 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
61	60	ad2antll 730	. . . 4 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
62		opelxpi 5661	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
63		opelxpi 5661	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
64	62, 63	anim12i 614	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
65	64	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
66	54	biimpar 477	. . . . . . 7 ⊢ ((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) → Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))
67	65, 66	jca 511	. . . . . 6 ⊢ ((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) → ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))) ∧ Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)))
68		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
69	68	sqxpeqd 5656	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑁) × (𝔼‘𝑁)))
70	69	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
71	69	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
72	70, 71	anbi12d 633	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ↔ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))))
73	72, 38	anbi12d 633	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)) ↔ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))) ∧ Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))))
74	73	rspcev 3565	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))) ∧ Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))) → ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)))
75	67, 74	sylan2 594	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))) → ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)))
76	75	exp32 420	. . . 4 ⊢ (𝑁 ∈ ℕ → (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) → ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)))))
77	61, 76	mpcom 38	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) → ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))))
78	59, 77	impbid 212	. 2 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2)) ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
79	28, 78	bitrid 283	1 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ⟨cop 4574   class class class wbr 5086   × cxp 5622  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  1c1 11030   − cmin 11368  ℕcn 12165  2c2 12227  ...cfz 13452  ↑cexp 14014  Σcsu 15639  𝔼cee 28970  Cgrccgr 28972

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-z 12516  df-uz 12780  df-fz 13453  df-seq 13955  df-sum 15640  df-ee 28973  df-cgr 28975

This theorem is referenced by:  axcgrrflx  28997  axcgrtr  28998  axcgrid  28999  axsegcon  29010  ax5seglem3  29014  ax5seglem6  29017  ax5seg  29021  axlowdimlem17  29041  ecgrtg  29066