Definition df-enr

Description: Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c , and is intended to be used only by the construction. From Proposition 9-4.1 of Gleason p. 126. (Contributed by NM, 25-Jul-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	df-enr	⊢ ~R = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ ( P × P ) ∧ 𝑦 ∈ ( P × P ) ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) ) ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cer	⊢ ~R
1		vx	⊢ 𝑥
2		vy	⊢ 𝑦
3	1	cv	⊢ 𝑥
4		cnp	⊢ P
5	4 4	cxp	⊢ ( P × P )
6	3 5	wcel	⊢ 𝑥 ∈ ( P × P )
7	2	cv	⊢ 𝑦
8	7 5	wcel	⊢ 𝑦 ∈ ( P × P )
9	6 8	wa	⊢ ( 𝑥 ∈ ( P × P ) ∧ 𝑦 ∈ ( P × P ) )
10		vz	⊢ 𝑧
11		vw	⊢ 𝑤
12		vv	⊢ 𝑣
13		vu	⊢ 𝑢
14	10	cv	⊢ 𝑧
15	11	cv	⊢ 𝑤
16	14 15	cop	⊢ ⟨ 𝑧 , 𝑤 ⟩
17	3 16	wceq	⊢ 𝑥 = ⟨ 𝑧 , 𝑤 ⟩
18	12	cv	⊢ 𝑣
19	13	cv	⊢ 𝑢
20	18 19	cop	⊢ ⟨ 𝑣 , 𝑢 ⟩
21	7 20	wceq	⊢ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩
22	17 21	wa	⊢ ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ )
23		cpp	⊢ +P
24	14 19 23	co	⊢ ( 𝑧 +P 𝑢 )
25	15 18 23	co	⊢ ( 𝑤 +P 𝑣 )
26	24 25	wceq	⊢ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 )
27	22 26	wa	⊢ ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) )
28	27 13	wex	⊢ ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) )
29	28 12	wex	⊢ ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) )
30	29 11	wex	⊢ ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) )
31	30 10	wex	⊢ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) )
32	9 31	wa	⊢ ( ( 𝑥 ∈ ( P × P ) ∧ 𝑦 ∈ ( P × P ) ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) ) )
33	32 1 2	copab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ ( P × P ) ∧ 𝑦 ∈ ( P × P ) ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) ) ) }
34	0 33	wceq	⊢ ~R = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ ( P × P ) ∧ 𝑦 ∈ ( P × P ) ) ∧ ∃ 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ( ( 𝑥 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝑦 = ⟨ 𝑣 , 𝑢 ⟩ ) ∧ ( 𝑧 +P 𝑢 ) = ( 𝑤 +P 𝑣 ) ) ) }