Theorem th3qlem2 6604

Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

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⟩)))

Assertion

Ref	Expression
th3qlem2	⊢ ((𝐴 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝐵 ∈ ((𝑆 × 𝑆) / ∼ )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))

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

Allowed substitution hints:   𝐴(𝑔,ℎ)   𝐵(𝑔,ℎ)

Proof of Theorem th3qlem2

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

Step	Hyp	Ref	Expression
1		th3q.2	. . 3 ⊢ ∼ Er (𝑆 × 𝑆)
2		eqid 2165	. . . . 5 ⊢ (𝑆 × 𝑆) = (𝑆 × 𝑆)
3		breq1 3985	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → (⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ↔ 𝑠 ∼ ⟨𝑢, 𝑡⟩))
4	3	anbi1d 461	. . . . . . 7 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) ↔ (𝑠 ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦)))
5		oveq1 5849	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → (⟨𝑤, 𝑣⟩ + 𝑥) = (𝑠 + 𝑥))
6	5	breq1d 3992	. . . . . . 7 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → ((⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦) ↔ (𝑠 + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦)))
7	4, 6	imbi12d 233	. . . . . 6 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → (((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦)) ↔ ((𝑠 ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦))))
8	7	imbi2d 229	. . . . 5 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → (((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦))) ↔ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) → ((𝑠 ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦)))))
9		breq2 3986	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → (𝑠 ∼ ⟨𝑢, 𝑡⟩ ↔ 𝑠 ∼ 𝑓))
10	9	anbi1d 461	. . . . . . 7 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → ((𝑠 ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) ↔ (𝑠 ∼ 𝑓 ∧ 𝑥 ∼ 𝑦)))
11		oveq1 5849	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → (⟨𝑢, 𝑡⟩ + 𝑦) = (𝑓 + 𝑦))
12	11	breq2d 3994	. . . . . . 7 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → ((𝑠 + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦) ↔ (𝑠 + 𝑥) ∼ (𝑓 + 𝑦)))
13	10, 12	imbi12d 233	. . . . . 6 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → (((𝑠 ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦)) ↔ ((𝑠 ∼ 𝑓 ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (𝑓 + 𝑦))))
14	13	imbi2d 229	. . . . 5 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → (((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) → ((𝑠 ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦))) ↔ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) → ((𝑠 ∼ 𝑓 ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (𝑓 + 𝑦)))))
15		breq1 3985	. . . . . . . . . 10 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → (⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩ ↔ 𝑥 ∼ ⟨𝑔, ℎ⟩))
16	15	anbi2d 460	. . . . . . . . 9 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) ↔ (⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ ⟨𝑔, ℎ⟩)))
17		oveq2 5850	. . . . . . . . . 10 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) = (⟨𝑤, 𝑣⟩ + 𝑥))
18	17	breq1d 3992	. . . . . . . . 9 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → ((⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩) ↔ (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)))
19	16, 18	imbi12d 233	. . . . . . . 8 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → (((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)) ↔ ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩))))
20	19	imbi2d 229	. . . . . . 7 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩))) ↔ (((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)))))
21		breq2 3986	. . . . . . . . . 10 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → (𝑥 ∼ ⟨𝑔, ℎ⟩ ↔ 𝑥 ∼ 𝑦))
22	21	anbi2d 460	. . . . . . . . 9 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ ⟨𝑔, ℎ⟩) ↔ (⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦)))
23		oveq2 5850	. . . . . . . . . 10 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩) = (⟨𝑢, 𝑡⟩ + 𝑦))
24	23	breq2d 3994	. . . . . . . . 9 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → ((⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩) ↔ (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦)))
25	22, 24	imbi12d 233	. . . . . . . 8 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → (((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)) ↔ ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦))))
26	25	imbi2d 229	. . . . . . 7 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩))) ↔ (((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦)))))
27		th3q.4	. . . . . . . 8 ⊢ ((((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⟩)))
28	27	expcom 115	. . . . . . 7 ⊢ (((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → (((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩))))
29	2, 20, 26, 28	2optocl 4681	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) → (((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦))))
30	29	com12 30	. . . . 5 ⊢ (((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) → (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦))))
31	2, 8, 14, 30	2optocl 4681	. . . 4 ⊢ ((𝑠 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)) → ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) → ((𝑠 ∼ 𝑓 ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (𝑓 + 𝑦))))
32	31	imp 123	. . 3 ⊢ (((𝑠 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)) ∧ (𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆))) → ((𝑠 ∼ 𝑓 ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (𝑓 + 𝑦)))
33	1, 32	th3qlem1 6603	. 2 ⊢ ((𝐴 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝐵 ∈ ((𝑆 × 𝑆) / ∼ )) → ∃*𝑧∃𝑠∃𝑥((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ∧ 𝑧 = [(𝑠 + 𝑥)] ∼ ))
34		vex 2729	. . . . . . 7 ⊢ 𝑤 ∈ V
35		vex 2729	. . . . . . 7 ⊢ 𝑣 ∈ V
36	34, 35	opex 4207	. . . . . 6 ⊢ ⟨𝑤, 𝑣⟩ ∈ V
37		vex 2729	. . . . . . 7 ⊢ 𝑢 ∈ V
38		vex 2729	. . . . . . 7 ⊢ 𝑡 ∈ V
39	37, 38	opex 4207	. . . . . 6 ⊢ ⟨𝑢, 𝑡⟩ ∈ V
40		eceq1 6536	. . . . . . . . 9 ⊢ (𝑠 = ⟨𝑤, 𝑣⟩ → [𝑠] ∼ = [⟨𝑤, 𝑣⟩] ∼ )
41	40	eqeq2d 2177	. . . . . . . 8 ⊢ (𝑠 = ⟨𝑤, 𝑣⟩ → (𝐴 = [𝑠] ∼ ↔ 𝐴 = [⟨𝑤, 𝑣⟩] ∼ ))
42		eceq1 6536	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑢, 𝑡⟩ → [𝑥] ∼ = [⟨𝑢, 𝑡⟩] ∼ )
43	42	eqeq2d 2177	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑡⟩ → (𝐵 = [𝑥] ∼ ↔ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ))
44	41, 43	bi2anan9 596	. . . . . . 7 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → ((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ↔ (𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ )))
45		oveq12 5851	. . . . . . . . 9 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → (𝑠 + 𝑥) = (⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩))
46	45	eceq1d 6537	. . . . . . . 8 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → [(𝑠 + 𝑥)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )
47	46	eqeq2d 2177	. . . . . . 7 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → (𝑧 = [(𝑠 + 𝑥)] ∼ ↔ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))
48	44, 47	anbi12d 465	. . . . . 6 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → (((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ∧ 𝑧 = [(𝑠 + 𝑥)] ∼ ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )))
49	36, 39, 48	spc2ev 2822	. . . . 5 ⊢ (((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) → ∃𝑠∃𝑥((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ∧ 𝑧 = [(𝑠 + 𝑥)] ∼ ))
50	49	exlimivv 1884	. . . 4 ⊢ (∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) → ∃𝑠∃𝑥((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ∧ 𝑧 = [(𝑠 + 𝑥)] ∼ ))
51	50	exlimivv 1884	. . 3 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) → ∃𝑠∃𝑥((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ∧ 𝑧 = [(𝑠 + 𝑥)] ∼ ))
52	51	moimi 2079	. 2 ⊢ (∃*𝑧∃𝑠∃𝑥((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ∧ 𝑧 = [(𝑠 + 𝑥)] ∼ ) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))
53	33, 52	syl 14	1 ⊢ ((𝐴 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝐵 ∈ ((𝑆 × 𝑆) / ∼ )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480  ∃*wmo 2015   ∈ wcel 2136  Vcvv 2726  ⟨cop 3579   class class class wbr 3982   × cxp 4602  (class class class)co 5842   Er wer 6498  [cec 6499   / cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fv 5196  df-ov 5845  df-er 6501  df-ec 6503  df-qs 6507

This theorem is referenced by:  th3qcor  6605  th3q  6606