Theorem brcgr 29058

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 5428	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5428	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eleq1 2849	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
4	3	anbi1d 640	. . . . 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 6862	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
6	5	fveq1d 6864	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)‘𝑖) = ((1st ‘⟨𝐴, 𝐵⟩)‘𝑖))
7		fveq2 6862	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
8	7	fveq1d 6864	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((2nd ‘𝑥)‘𝑖) = ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))
9	6, 8	oveq12d 7409	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖)) = (((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖)))
10	9	oveq1d 7406	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = ((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2))
11	10	sumeq2sdv 15721	. . . . . 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 2763	. . . . 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 641	. . . 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 3185	. . 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 2849	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
16	15	anbi2d 639	. . . . 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 6862	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
18	17	fveq1d 6864	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((1st ‘𝑦)‘𝑖) = ((1st ‘⟨𝐶, 𝐷⟩)‘𝑖))
19		fveq2 6862	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
20	19	fveq1d 6864	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((2nd ‘𝑦)‘𝑖) = ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))
21	18, 20	oveq12d 7409	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖)) = (((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖)))
22	21	oveq1d 7406	. . . . . . 7 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2) = ((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))
23	22	sumeq2sdv 15721	. . . . . 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 2772	. . . . 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 641	. . . 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 3185	. . 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 29050	. . 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 5510	. 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 5686	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → 𝐷 ∈ (𝔼‘𝑛))
30	29	ad2antll 739	. . . . . . . . . 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 787	. . . . . . . . . 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 29056	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
33	30, 31, 32	syl2anc 593	. . . . . . . . 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 725	. . . . . . . 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 7399	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
36	35	sumeq1d 15718	. . . . . . . . 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 15718	. . . . . . . . 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 2777	. . . . . . . 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 7977	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
41	40	fveq1d 6864	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) = (𝐴‘𝑖))
42		op2ndg 7978	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
43	42	fveq1d 6864	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖) = (𝐵‘𝑖))
44	41, 43	oveq12d 7409	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖)) = ((𝐴‘𝑖) − (𝐵‘𝑖)))
45	44	oveq1d 7406	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = (((𝐴‘𝑖) − (𝐵‘𝑖))↑2))
46	45	sumeq2sdv 15721	. . . . . . . . 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 7977	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
48	47	fveq1d 6864	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) = (𝐶‘𝑖))
49		op2ndg 7978	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
50	49	fveq1d 6864	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖) = (𝐷‘𝑖))
51	48, 50	oveq12d 7409	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖)) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
52	51	oveq1d 7406	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) = (((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
53	52	sumeq2sdv 15721	. . . . . . . . 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 2775	. . . . . . . 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 736	. . . . . . 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 281	. . . . . 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 231	. . . . 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 457	. . . 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 3162	. . 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 29054	. . . . 5 ⊢ (𝐷 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
61	60	ad2antll 739	. . . 4 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
62		opelxpi 5680	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
63		opelxpi 5680	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
64	62, 63	anim12i 622	. . . . . . . 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 484	. . . . . . 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 481	. . . . . . 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 519	. . . . . 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 6862	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
69	68	sqxpeqd 5675	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑁) × (𝔼‘𝑁)))
70	69	eleq2d 2847	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
71	69	eleq2d 2847	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
72	70, 71	anbi12d 641	. . . . . . . 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 641	. . . . . . 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 3580	. . . . . 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 602	. . . . 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 424	. . . 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 214	. 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 285	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 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ⟨cop 4585   class class class wbr 5097   × cxp 5641  ‘cfv 6516  (class class class)co 7391  1^st c1st 7963  2^nd c2nd 7964  1c1 11068   − cmin 11408  ℕcn 12204  2c2 12266  ...cfz 13506  ↑cexp 14068  Σcsu 15704  𝔼cee 29045  Cgrccgr 29047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-er 8672  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-z 12563  df-uz 12834  df-fz 13507  df-seq 14009  df-sum 15705  df-ee 29048  df-cgr 29050

This theorem is referenced by:  axcgrrflx  29072  axcgrtr  29073  axcgrid  29074  axsegcon  29085  ax5seglem3  29089  ax5seglem6  29092  ax5seg  29096  axlowdimlem17  29116  ecgrtg  29141