Theorem th3qlem2 6232

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 2081	. . . . 5 ⊢ (𝑆 × 𝑆) = (𝑆 × 𝑆)
3		breq1 3788	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → (⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ↔ 𝑠 ∼ ⟨𝑢, 𝑡⟩))
4	3	anbi1d 452	. . . . . . 7 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) ↔ (𝑠 ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦)))
5		oveq1 5539	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → (⟨𝑤, 𝑣⟩ + 𝑥) = (𝑠 + 𝑥))
6	5	breq1d 3795	. . . . . . 7 ⊢ (⟨𝑤, 𝑣⟩ = 𝑠 → ((⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦) ↔ (𝑠 + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦)))
7	4, 6	imbi12d 232	. . . . . 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 228	. . . . 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 3789	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → (𝑠 ∼ ⟨𝑢, 𝑡⟩ ↔ 𝑠 ∼ 𝑓))
10	9	anbi1d 452	. . . . . . 7 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → ((𝑠 ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦) ↔ (𝑠 ∼ 𝑓 ∧ 𝑥 ∼ 𝑦)))
11		oveq1 5539	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → (⟨𝑢, 𝑡⟩ + 𝑦) = (𝑓 + 𝑦))
12	11	breq2d 3797	. . . . . . 7 ⊢ (⟨𝑢, 𝑡⟩ = 𝑓 → ((𝑠 + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦) ↔ (𝑠 + 𝑥) ∼ (𝑓 + 𝑦)))
13	10, 12	imbi12d 232	. . . . . 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 228	. . . . 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 3788	. . . . . . . . . 10 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → (⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩ ↔ 𝑥 ∼ ⟨𝑔, ℎ⟩))
16	15	anbi2d 451	. . . . . . . . 9 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓⟩ ∼ ⟨𝑔, ℎ⟩) ↔ (⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ ⟨𝑔, ℎ⟩)))
17		oveq2 5540	. . . . . . . . . 10 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → (⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) = (⟨𝑤, 𝑣⟩ + 𝑥))
18	17	breq1d 3795	. . . . . . . . 9 ⊢ (⟨𝑠, 𝑓⟩ = 𝑥 → ((⟨𝑤, 𝑣⟩ + ⟨𝑠, 𝑓⟩) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩) ↔ (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩)))
19	16, 18	imbi12d 232	. . . . . . . 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 228	. . . . . . 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 3789	. . . . . . . . . 10 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → (𝑥 ∼ ⟨𝑔, ℎ⟩ ↔ 𝑥 ∼ 𝑦))
22	21	anbi2d 451	. . . . . . . . 9 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → ((⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ ⟨𝑔, ℎ⟩) ↔ (⟨𝑤, 𝑣⟩ ∼ ⟨𝑢, 𝑡⟩ ∧ 𝑥 ∼ 𝑦)))
23		oveq2 5540	. . . . . . . . . 10 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩) = (⟨𝑢, 𝑡⟩ + 𝑦))
24	23	breq2d 3797	. . . . . . . . 9 ⊢ (⟨𝑔, ℎ⟩ = 𝑦 → ((⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + ⟨𝑔, ℎ⟩) ↔ (⟨𝑤, 𝑣⟩ + 𝑥) ∼ (⟨𝑢, 𝑡⟩ + 𝑦)))
25	22, 24	imbi12d 232	. . . . . . . 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 228	. . . . . . 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 114	. . . . . . 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 4435	. . . . . 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 4435	. . . 4 ⊢ ((𝑠 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)) → ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) → ((𝑠 ∼ 𝑓 ∧ 𝑥 ∼ 𝑦) → (𝑠 + 𝑥) ∼ (𝑓 + 𝑦))))
32	31	imp 122	. . 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 6231	. 2 ⊢ ((𝐴 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝐵 ∈ ((𝑆 × 𝑆) / ∼ )) → ∃*𝑧∃𝑠∃𝑥((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ∧ 𝑧 = [(𝑠 + 𝑥)] ∼ ))
34		vex 2604	. . . . . . 7 ⊢ 𝑤 ∈ V
35		vex 2604	. . . . . . 7 ⊢ 𝑣 ∈ V
36	34, 35	opex 3984	. . . . . 6 ⊢ ⟨𝑤, 𝑣⟩ ∈ V
37		vex 2604	. . . . . . 7 ⊢ 𝑢 ∈ V
38		vex 2604	. . . . . . 7 ⊢ 𝑡 ∈ V
39	37, 38	opex 3984	. . . . . 6 ⊢ ⟨𝑢, 𝑡⟩ ∈ V
40		eceq1 6164	. . . . . . . . 9 ⊢ (𝑠 = ⟨𝑤, 𝑣⟩ → [𝑠] ∼ = [⟨𝑤, 𝑣⟩] ∼ )
41	40	eqeq2d 2092	. . . . . . . 8 ⊢ (𝑠 = ⟨𝑤, 𝑣⟩ → (𝐴 = [𝑠] ∼ ↔ 𝐴 = [⟨𝑤, 𝑣⟩] ∼ ))
42		eceq1 6164	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑢, 𝑡⟩ → [𝑥] ∼ = [⟨𝑢, 𝑡⟩] ∼ )
43	42	eqeq2d 2092	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑡⟩ → (𝐵 = [𝑥] ∼ ↔ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ))
44	41, 43	bi2anan9 570	. . . . . . 7 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → ((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ↔ (𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ )))
45		oveq12 5541	. . . . . . . . 9 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → (𝑠 + 𝑥) = (⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩))
46	45	eceq1d 6165	. . . . . . . 8 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → [(𝑠 + 𝑥)] ∼ = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ )
47	46	eqeq2d 2092	. . . . . . 7 ⊢ ((𝑠 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = ⟨𝑢, 𝑡⟩) → (𝑧 = [(𝑠 + 𝑥)] ∼ ↔ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ))
48	44, 47	anbi12d 456	. . . . . 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 2693	. . . . 5 ⊢ (((𝐴 = [⟨𝑤, 𝑣⟩] ∼ ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ∼ ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ + ⟨𝑢, 𝑡⟩)] ∼ ) → ∃𝑠∃𝑥((𝐴 = [𝑠] ∼ ∧ 𝐵 = [𝑥] ∼ ) ∧ 𝑧 = [(𝑠 + 𝑥)] ∼ ))
50	49	exlimivv 1817	. . . 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 1817	. . 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 2006	. 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 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃*wmo 1942  Vcvv 2601  ⟨cop 3401   class class class wbr 3785   × cxp 4361  (class class class)co 5532   Er wer 6126  [cec 6127   / cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fv 4930  df-ov 5535  df-er 6129  df-ec 6131  df-qs 6135

This theorem is referenced by:  th3qcor  6233  th3q  6234