Theorem brcgr 26686

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 5356	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5356	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eleq1 2900	. . . . . 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 6670	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
6	5	fveq1d 6672	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((1st ‘𝑥)‘𝑖) = ((1st ‘⟨𝐴, 𝐵⟩)‘𝑖))
7		fveq2 6670	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
8	7	fveq1d 6672	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((2nd ‘𝑥)‘𝑖) = ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))
9	6, 8	oveq12d 7174	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖)) = (((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖)))
10	9	oveq1d 7171	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = ((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2))
11	10	sumeq2sdv 15061	. . . . . 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 2823	. . . . 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 3297	. . 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 2900	. . . . . 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 6670	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
18	17	fveq1d 6672	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((1st ‘𝑦)‘𝑖) = ((1st ‘⟨𝐶, 𝐷⟩)‘𝑖))
19		fveq2 6670	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
20	19	fveq1d 6672	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((2nd ‘𝑦)‘𝑖) = ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))
21	18, 20	oveq12d 7174	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖)) = (((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖)))
22	21	oveq1d 7171	. . . . . . 7 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2) = ((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2))
23	22	sumeq2sdv 15061	. . . . . 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 2832	. . . . 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 3297	. . 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 26679	. . 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 5430	. 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 5597	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → 𝐷 ∈ (𝔼‘𝑛))
30	29	ad2antll 727	. . . . . . . . . 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 776	. . . . . . . . . 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 26684	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
33	30, 31, 32	syl2anc 586	. . . . . . . . 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 713	. . . . . . . 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 7164	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
36	35	sumeq1d 15058	. . . . . . . . 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 15058	. . . . . . . . 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 2837	. . . . . . . 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 7701	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
41	40	fveq1d 6672	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) = (𝐴‘𝑖))
42		op2ndg 7702	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
43	42	fveq1d 6672	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖) = (𝐵‘𝑖))
44	41, 43	oveq12d 7174	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖)) = ((𝐴‘𝑖) − (𝐵‘𝑖)))
45	44	oveq1d 7171	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ((((1st ‘⟨𝐴, 𝐵⟩)‘𝑖) − ((2nd ‘⟨𝐴, 𝐵⟩)‘𝑖))↑2) = (((𝐴‘𝑖) − (𝐵‘𝑖))↑2))
46	45	sumeq2sdv 15061	. . . . . . . . 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 7701	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
48	47	fveq1d 6672	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) = (𝐶‘𝑖))
49		op2ndg 7702	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
50	49	fveq1d 6672	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖) = (𝐷‘𝑖))
51	48, 50	oveq12d 7174	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖)) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
52	51	oveq1d 7171	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((((1st ‘⟨𝐶, 𝐷⟩)‘𝑖) − ((2nd ‘⟨𝐶, 𝐷⟩)‘𝑖))↑2) = (((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
53	52	sumeq2sdv 15061	. . . . . . . . 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 2838	. . . . . . . 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 724	. . . . . . 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 456	. . . 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 3284	. . 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 26682	. . . . 5 ⊢ (𝐷 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
61	60	ad2antll 727	. . . 4 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
62		opelxpi 5592	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
63		opelxpi 5592	. . . . . . . . 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 483	. . . . . . 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 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)) → 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 514	. . . . . 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 6670	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
69	68	sqxpeqd 5587	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑁) × (𝔼‘𝑁)))
70	69	eleq2d 2898	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
71	69	eleq2d 2898	. . . . . . . . 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 3623	. . . . . 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 423	. . . 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	syl5bb 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 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  ⟨cop 4573   class class class wbr 5066   × cxp 5553  ‘cfv 6355  (class class class)co 7156  1^st c1st 7687  2^nd c2nd 7688  1c1 10538   − cmin 10870  ℕcn 11638  2c2 11693  ...cfz 12893  ↑cexp 13430  Σcsu 15042  𝔼cee 26674  Cgrccgr 26676

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-seq 13371  df-sum 15043  df-ee 26677  df-cgr 26679

This theorem is referenced by:  axcgrrflx  26700  axcgrtr  26701  axcgrid  26702  axsegcon  26713  ax5seglem3  26717  ax5seglem6  26720  ax5seg  26724  axlowdimlem17  26744  ecgrtg  26769