Theorem th3q 6658

Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
th3q.1	⊢ ∼ ∈ V
th3q.2	⊢ ∼ Er (𝑆 × 𝑆)
th3q.4	⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)))
th3q.5	⊢ 𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))}

Assertion

Ref	Expression
th3q	⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ 𝐺[⟨𝐶, 𝐷⟩] ∼ ) = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ )

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓,𝑔,ℎ, ∼   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓,𝑔,ℎ   𝑥,𝐴,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡   𝑥,𝐷,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡   𝑥, + ,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓,𝑔,ℎ

Allowed substitution hints:   𝐴(𝑔,ℎ)   𝐵(𝑔,ℎ)   𝐶(𝑓,𝑔,ℎ,𝑠)   𝐷(𝑓,𝑔,ℎ,𝑠)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,ℎ,𝑠)

Proof of Theorem th3q

Step	Hyp	Ref	Expression
1		opelxpi 4673	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2		th3q.1	. . . . 5 ⊢ ∼ ∈ V
3	2	ecelqsi 6607	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → [⟨𝐴, 𝐵⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ ))
4	1, 3	syl 14	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → [⟨𝐴, 𝐵⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ ))
5		opelxpi 4673	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
6	2	ecelqsi 6607	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) → [⟨𝐶, 𝐷⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ ))
7	5, 6	syl 14	. . 3 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → [⟨𝐶, 𝐷⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ ))
8	4, 7	anim12i 338	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ ) ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ )))
9		eqid 2189	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼
10		eqid 2189	. . . 4 ⊢ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼
11	9, 10	pm3.2i 272	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )
12		eqid 2189	. . 3 ⊢ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼
13		opeq12 3795	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14		eceq1 6588	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ )
15	14	eqeq2d 2201	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ↔ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ ))
16	15	anbi1d 465	. . . . . . 7 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ↔ ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )))
17		oveq1 5898	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩) = (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩))
18	17	eceq1d 6589	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ )
19	18	eqeq2d 2201	. . . . . . 7 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ ↔ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ))
20	16, 19	anbi12d 473	. . . . . 6 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ((([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) ↔ (([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ )))
21	13, 20	syl 14	. . . . 5 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) ↔ (([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ )))
22	21	spc2egv 2842	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) → ∃𝑤∃𝑣(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ )))
23		opeq12 3795	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
24		eceq1 6588	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )
25	24	eqeq2d 2201	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ↔ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ))
26	25	anbi2d 464	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ↔ ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )))
27		oveq2 5899	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩) = (⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩))
28	27	eceq1d 6589	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ )
29	28	eqeq2d 2201	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ↔ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ ))
30	26, 29	anbi12d 473	. . . . . . 7 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ((([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) ↔ (([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ )))
31	23, 30	syl 14	. . . . . 6 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) ↔ (([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ )))
32	31	spc2egv 2842	. . . . 5 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → ((([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) → ∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )))
33	32	2eximdv 1893	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (∃𝑤∃𝑣(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )))
34	22, 33	sylan9 409	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ((([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )))
35	11, 12, 34	mp2ani 432	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))
36		ecexg 6557	. . . 4 ⊢ ( ∼ ∈ V → [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ∈ V)
37	2, 36	ax-mp 5	. . 3 ⊢ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ∈ V
38		eqeq1 2196	. . . . . . . 8 ⊢ (𝑥 = [⟨𝐴, 𝐵⟩] ∼ → (𝑥 = [⟨𝑤, 𝑣⟩] ∼ ↔ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ))
39		eqeq1 2196	. . . . . . . 8 ⊢ (𝑦 = [⟨𝐶, 𝐷⟩] ∼ → (𝑦 = [⟨𝑢, 𝑡⟩] ∼ ↔ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ))
40	38, 39	bi2anan9 606	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ∼ ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ∼ ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ↔ ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ )))
41		eqeq1 2196	. . . . . . 7 ⊢ (𝑧 = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ → (𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ↔ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))
42	40, 41	bi2anan9 606	. . . . . 6 ⊢ (((𝑥 = [⟨𝐴, 𝐵⟩] ∼ ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ∼ ) ∧ 𝑧 = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) ↔ (([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )))
43	42	3impa 1196	. . . . 5 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ∼ ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ∼ ∧ 𝑧 = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) ↔ (([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )))
44	43	4exbidv 1881	. . . 4 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ∼ ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ∼ ∧ 𝑧 = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )))
45		th3q.2	. . . . 5 ⊢ ∼ Er (𝑆 × 𝑆)
46		th3q.4	. . . . 5 ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)))
47	2, 45, 46	th3qlem2 6656	. . . 4 ⊢ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))
48		th3q.5	. . . 4 ⊢ 𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))}
49	44, 47, 48	ovig 6013	. . 3 ⊢ (([⟨𝐴, 𝐵⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ ) ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ∈ V) → (∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) → ([⟨𝐴, 𝐵⟩] ∼ 𝐺[⟨𝐶, 𝐷⟩] ∼ ) = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ))
50	37, 49	mp3an3 1337	. 2 ⊢ (([⟨𝐴, 𝐵⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ ) ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ ((𝑆 × 𝑆) / ∼ )) → (∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑤, 𝑣⟩] ∼ ∧ [⟨𝐶, 𝐷⟩] ∼ = [⟨𝑢, 𝑡⟩] ∼ ) ∧ [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) → ([⟨𝐴, 𝐵⟩] ∼ 𝐺[⟨𝐶, 𝐷⟩] ∼ ) = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ ))
51	8, 35, 50	sylc 62	1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ 𝐺[⟨𝐶, 𝐷⟩] ∼ ) = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1503   ∈ wcel 2160  Vcvv 2752  ⟨cop 3610   class class class wbr 4018   × cxp 4639  (class class class)co 5891  {coprab 5892   Er wer 6550  [cec 6551   / cqs 6552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-er 6553  df-ec 6555  df-qs 6559

This theorem is referenced by:  oviec  6659