Theorem dfplpq2 7295

Description: Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)

Assertion

Ref	Expression
dfplpq2	⊢ +pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))}

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Proof of Theorem dfplpq2

Step	Hyp	Ref	Expression
1		df-mpo 5847	. 2 ⊢ (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)}
2		df-plpq 7285	. 2 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
3		1st2nd2 6143	. . . . . . . . . 10 ⊢ (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
4	3	eqeq1d 2174	. . . . . . . . 9 ⊢ (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5		1st2nd2 6143	. . . . . . . . . 10 ⊢ (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
6	5	eqeq1d 2174	. . . . . . . . 9 ⊢ (𝑦 ∈ (N × N) → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩))
7	4, 6	bi2anan9 596	. . . . . . . 8 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩)))
8	7	anbi1d 461	. . . . . . 7 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩)))
9		xp1st 6133	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
10	9	ad2antlr 481	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (1st ‘𝑦) ∈ N)
11	7	biimpa 294	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩))
12	11	simprd 113	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩)
13		vex 2729	. . . . . . . . . . . . . . . . 17 ⊢ 𝑢 ∈ V
14		vex 2729	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
15	13, 14	opth2 4218	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩ ↔ ((1st ‘𝑦) = 𝑢 ∧ (2nd ‘𝑦) = 𝑓))
16	15	simplbi 272	. . . . . . . . . . . . . . 15 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩ → (1st ‘𝑦) = 𝑢)
17	16	eleq1d 2235	. . . . . . . . . . . . . 14 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩ → ((1st ‘𝑦) ∈ N ↔ 𝑢 ∈ N))
18	12, 17	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ((1st ‘𝑦) ∈ N ↔ 𝑢 ∈ N))
19	10, 18	mpbid 146	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → 𝑢 ∈ N)
20		xp2nd 6134	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
21	20	ad2antrr 480	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (2nd ‘𝑥) ∈ N)
22	11	simpld 111	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩)
23		vex 2729	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
24		vex 2729	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ V
25	23, 24	opth2 4218	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ↔ ((1st ‘𝑥) = 𝑤 ∧ (2nd ‘𝑥) = 𝑣))
26	25	simprbi 273	. . . . . . . . . . . . . . 15 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ → (2nd ‘𝑥) = 𝑣)
27	26	eleq1d 2235	. . . . . . . . . . . . . 14 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ → ((2nd ‘𝑥) ∈ N ↔ 𝑣 ∈ N))
28	22, 27	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ((2nd ‘𝑥) ∈ N ↔ 𝑣 ∈ N))
29	21, 28	mpbid 146	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → 𝑣 ∈ N)
30		mulcompig 7272	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
31	19, 29, 30	syl2anc 409	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
32	31	oveq2d 5858	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)) = ((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)))
33	32	opeq1d 3764	. . . . . . . . 9 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩ = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)
34	33	eqeq2d 2177	. . . . . . . 8 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩ ↔ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))
35	34	pm5.32da 448	. . . . . . 7 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)))
36	8, 35	bitr3d 189	. . . . . 6 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)))
37	36	4exbidv 1858	. . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤∃𝑣∃𝑢∃𝑓((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)))
38		xp1st 6133	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
39	38, 20	jca 304	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → ((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑥) ∈ N))
40		xp2nd 6134	. . . . . . 7 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
41	9, 40	jca 304	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → ((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑦) ∈ N))
42		simpll 519	. . . . . . . . . . 11 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑤 = (1st ‘𝑥))
43		simprr 522	. . . . . . . . . . 11 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑓 = (2nd ‘𝑦))
44	42, 43	oveq12d 5860	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑤 ·N 𝑓) = ((1st ‘𝑥) ·N (2nd ‘𝑦)))
45		simprl 521	. . . . . . . . . . 11 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑢 = (1st ‘𝑦))
46		simplr 520	. . . . . . . . . . 11 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑣 = (2nd ‘𝑥))
47	45, 46	oveq12d 5860	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑢 ·N 𝑣) = ((1st ‘𝑦) ·N (2nd ‘𝑥)))
48	44, 47	oveq12d 5860	. . . . . . . . 9 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → ((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)) = (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
49	46, 43	oveq12d 5860	. . . . . . . . 9 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑣 ·N 𝑓) = ((2nd ‘𝑥) ·N (2nd ‘𝑦)))
50	48, 49	opeq12d 3766	. . . . . . . 8 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩ = ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
51	50	eqeq2d 2177	. . . . . . 7 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩ ↔ 𝑧 = ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
52	51	copsex4g 4225	. . . . . 6 ⊢ ((((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑥) ∈ N) ∧ ((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑦) ∈ N)) → (∃𝑤∃𝑣∃𝑢∃𝑓((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
53	39, 41, 52	syl2an 287	. . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤∃𝑣∃𝑢∃𝑓((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑢 ·N 𝑣)), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
54	37, 53	bitr3d 189	. . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩) ↔ 𝑧 = ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
55	54	pm5.32i 450	. . 3 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩)) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩))
56	55	oprabbii 5897	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ 𝑧 = ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)}
57	1, 2, 56	3eqtr4i 2196	1 ⊢ +pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))}

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ⟨cop 3579   × cxp 4602  ‘cfv 5188  (class class class)co 5842  {coprab 5843   ∈ cmpo 5844  1^st c1st 6106  2^nd c2nd 6107  Ncnpi 7213   +_N cpli 7214   ·_N cmi 7215   +_pQ cplpq 7217

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-in1 604  ax-in2 605  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-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  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-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  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-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389  df-ni 7245  df-mi 7247  df-plpq 7285

This theorem is referenced by:  addpipqqs  7311