Theorem dfplpq2 7063

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 5711	. 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 7053	. 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 6003	. . . . . . . . . 10 ⊢ (𝑥 ∈ (N × N) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
4	3	eqeq1d 2108	. . . . . . . . 9 ⊢ (𝑥 ∈ (N × N) → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩))
5		1st2nd2 6003	. . . . . . . . . 10 ⊢ (𝑦 ∈ (N × N) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
6	5	eqeq1d 2108	. . . . . . . . 9 ⊢ (𝑦 ∈ (N × N) → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩))
7	4, 6	bi2anan9 576	. . . . . . . 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 456	. . . . . . 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 5994	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
10	9	ad2antlr 476	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (1st ‘𝑦) ∈ N)
11	7	biimpa 292	. . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . . . 17 ⊢ 𝑢 ∈ V
14		vex 2644	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
15	13, 14	opth2 4100	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩ ↔ ((1st ‘𝑦) = 𝑢 ∧ (2nd ‘𝑦) = 𝑓))
16	15	simplbi 270	. . . . . . . . . . . . . . 15 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑢, 𝑓⟩ → (1st ‘𝑦) = 𝑢)
17	16	eleq1d 2168	. . . . . . . . . . . . . 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 5995	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
21	20	ad2antrr 475	. . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
24		vex 2644	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ V
25	23, 24	opth2 4100	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ ↔ ((1st ‘𝑥) = 𝑤 ∧ (2nd ‘𝑥) = 𝑣))
26	25	simprbi 271	. . . . . . . . . . . . . . 15 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑤, 𝑣⟩ → (2nd ‘𝑥) = 𝑣)
27	26	eleq1d 2168	. . . . . . . . . . . . . 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 7040	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
31	19, 29, 30	syl2anc 406	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
32	31	oveq2d 5722	. . . . . . . . . 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 3658	. . . . . . . . 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 2111	. . . . . . . 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 443	. . . . . . 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 1809	. . . . 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 5994	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
39	38, 20	jca 302	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → ((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑥) ∈ N))
40		xp2nd 5995	. . . . . . 7 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
41	9, 40	jca 302	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → ((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑦) ∈ N))
42		simpll 499	. . . . . . . . . . 11 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑤 = (1st ‘𝑥))
43		simprr 502	. . . . . . . . . . 11 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑓 = (2nd ‘𝑦))
44	42, 43	oveq12d 5724	. . . . . . . . . 10 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → (𝑤 ·N 𝑓) = ((1st ‘𝑥) ·N (2nd ‘𝑦)))
45		simprl 501	. . . . . . . . . . 11 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑢 = (1st ‘𝑦))
46		simplr 500	. . . . . . . . . . 11 ⊢ (((𝑤 = (1st ‘𝑥) ∧ 𝑣 = (2nd ‘𝑥)) ∧ (𝑢 = (1st ‘𝑦) ∧ 𝑓 = (2nd ‘𝑦))) → 𝑣 = (2nd ‘𝑥))
47	45, 46	oveq12d 5724	. . . . . . . . . 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 5724	. . . . . . . . 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 5724	. . . . . . . . 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 3660	. . . . . . . 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 2111	. . . . . . 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 4107	. . . . . 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 285	. . . . 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 445	. . 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 5758	. 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 2130	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 1299  ∃wex 1436   ∈ wcel 1448  ⟨cop 3477   × cxp 4475  ‘cfv 5059  (class class class)co 5706  {coprab 5707   ∈ cmpo 5708  1^st c1st 5967  2^nd c2nd 5968  Ncnpi 6981   +_N cpli 6982   ·_N cmi 6983   +_pQ cplpq 6985

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-oadd 6247  df-omul 6248  df-ni 7013  df-mi 7015  df-plpq 7053

This theorem is referenced by:  addpipqqs  7079