Theorem brcgr 28873

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 5399	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5399	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eleq1 2819	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
4	3	anbi1d 631	. . . . 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 6817	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
6	5	fveq1d 6819	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)‘𝑖) = ((1st ‘⟨𝐴, 𝐵⟩)‘𝑖))
7		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
8	7	fveq1d 6819	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((2nd ‘𝑥)‘𝑖) = ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))
9	6, 8	oveq12d 7359	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖)) = (((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖)))
10	9	oveq1d 7356	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = ((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2))
11	10	sumeq2sdv 15605	. . . . . 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 2733	. . . . 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 632	. . . 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 3156	. . 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 2819	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
16	15	anbi2d 630	. . . . 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 6817	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
18	17	fveq1d 6819	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((1st ‘𝑦)‘𝑖) = ((1st ‘⟨𝐶, 𝐷⟩)‘𝑖))
19		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
20	19	fveq1d 6819	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((2nd ‘𝑦)‘𝑖) = ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))
21	18, 20	oveq12d 7359	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖)) = (((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖)))
22	21	oveq1d 7356	. . . . . . 7 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2) = ((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))
23	22	sumeq2sdv 15605	. . . . . 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 2742	. . . . 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 632	. . . 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 3156	. . 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 28866	. . 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 5478	. 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 5654	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → 𝐷 ∈ (𝔼‘𝑛))
30	29	ad2antll 729	. . . . . . . . . 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 777	. . . . . . . . . 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 28871	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
33	30, 31, 32	syl2anc 584	. . . . . . . . 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 715	. . . . . . . 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 7349	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
36	35	sumeq1d 15602	. . . . . . . . 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 15602	. . . . . . . . 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 2747	. . . . . . . 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 7928	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
41	40	fveq1d 6819	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) = (𝐴‘𝑖))
42		op2ndg 7929	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
43	42	fveq1d 6819	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖) = (𝐵‘𝑖))
44	41, 43	oveq12d 7359	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖)) = ((𝐴‘𝑖) − (𝐵‘𝑖)))
45	44	oveq1d 7356	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = (((𝐴‘𝑖) − (𝐵‘𝑖))↑2))
46	45	sumeq2sdv 15605	. . . . . . . . 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 7928	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
48	47	fveq1d 6819	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) = (𝐶‘𝑖))
49		op2ndg 7929	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
50	49	fveq1d 6819	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖) = (𝐷‘𝑖))
51	48, 50	oveq12d 7359	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖)) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
52	51	oveq1d 7356	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) = (((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
53	52	sumeq2sdv 15605	. . . . . . . . 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 2745	. . . . . . . 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 726	. . . . . . 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 3133	. . 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 28869	. . . . 5 ⊢ (𝐷 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
61	60	ad2antll 729	. . . 4 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
62		opelxpi 5648	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
63		opelxpi 5648	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
64	62, 63	anim12i 613	. . . . . . . 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 6817	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
69	68	sqxpeqd 5643	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑁) × (𝔼‘𝑁)))
70	69	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
71	69	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
72	70, 71	anbi12d 632	. . . . . . . 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 632	. . . . . . 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 3572	. . . . . 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 593	. . . . 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 1541   ∈ wcel 2111  ∃wrex 3056  ⟨cop 4577   class class class wbr 5086   × cxp 5609  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  1c1 11002   − cmin 11339  ℕcn 12120  2c2 12175  ...cfz 13402  ↑cexp 13963  Σcsu 15588  𝔼cee 28861  Cgrccgr 28863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-z 12464  df-uz 12728  df-fz 13403  df-seq 13904  df-sum 15589  df-ee 28864  df-cgr 28866

This theorem is referenced by:  axcgrrflx  28887  axcgrtr  28888  axcgrid  28889  axsegcon  28900  ax5seglem3  28904  ax5seglem6  28907  ax5seg  28911  axlowdimlem17  28931  ecgrtg  28956