Theorem addsrpr 11163

Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
addsrpr	⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )

Proof of Theorem addsrpr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxpi 5687	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ⟨𝐴, 𝐵⟩ ∈ (P × P))
2		enrex 11155	. . . . 5 ⊢ ~R ∈ V
3	2	ecelqsi 8793	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (P × P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
4	1, 3	syl 18	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
5		opelxpi 5687	. . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → ⟨𝐶, 𝐷⟩ ∈ (P × P))
6	2	ecelqsi 8793	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ ∈ (P × P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
7	5, 6	syl 18	. . 3 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
8	4, 7	anim12i 625	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )))
9		eqid 2760	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R
10		eqid 2760	. . . 4 ⊢ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R
11	9, 10	pm3.2i 476	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
12		eqid 2760	. . 3 ⊢ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R
13		opeq12 4841	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14	13	eceq1d 8761	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝐴, 𝐵⟩] ~R )
15	14	eqeq2d 2771	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ))
16	15	anbi1d 643	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ↔ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )))
17		simpl 488	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑤 = 𝐴)
18	17	oveq1d 7435	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 +P 𝐶) = (𝐴 +P 𝐶))
19		simpr 490	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
20	19	oveq1d 7435	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 +P 𝐷) = (𝐵 +P 𝐷))
21	18, 20	opeq12d 4847	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩ = ⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩)
22	21	eceq1d 8761	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
23	22	eqeq2d 2771	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ([⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R ↔ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ))
24	16, 23	anbi12d 644	. . . . 5 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R ) ↔ (([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )))
25	24	spc2egv 3553	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → ∃𝑤∃𝑣(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R )))
26		opeq12 4841	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
27	26	eceq1d 8761	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
28	27	eqeq2d 2771	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
29	28	anbi2d 642	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ↔ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )))
30		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑢 = 𝐶)
31	30	oveq2d 7436	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 +P 𝑢) = (𝑤 +P 𝐶))
32		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑡 = 𝐷)
33	32	oveq2d 7436	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 +P 𝑡) = (𝑣 +P 𝐷))
34	31, 33	opeq12d 4847	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩ = ⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩)
35	34	eceq1d 8761	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R )
36	35	eqeq2d 2771	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ↔ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R ))
37	29, 36	anbi12d 644	. . . . . 6 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ (([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R )))
38	37	spc2egv 3553	. . . . 5 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R ) → ∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
39	38	2eximdv 1951	. . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → (∃𝑤∃𝑣(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
40	25, 39	sylan9 517	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
41	11, 12, 40	mp2ani 711	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
42		ecexg 8724	. . . 4 ⊢ ( ~R ∈ V → [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ∈ V)
43	2, 42	ax-mp 5	. . 3 ⊢ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ∈ V
44		simp1 1154	. . . . . . . 8 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → 𝑥 = [⟨𝐴, 𝐵⟩] ~R )
45	44	eqeq1d 2762	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (𝑥 = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ))
46		simp2 1155	. . . . . . . 8 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → 𝑦 = [⟨𝐶, 𝐷⟩] ~R )
47	46	eqeq1d 2762	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (𝑦 = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ))
48	45, 47	anbi12d 644	. . . . . 6 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ↔ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R )))
49		simp3 1156	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
50	49	eqeq1d 2762	. . . . . 6 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ↔ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
51	48, 50	anbi12d 644	. . . . 5 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ (([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
52	51	4exbidv 1958	. . . 4 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
53		addsrmo 11161	. . . 4 ⊢ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
54		df-plr 11145	. . . . 5 ⊢ +R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))}
55		df-nr 11144	. . . . . . . . 9 ⊢ R = ((P × P) / ~R )
56	55	eleq2i 2852	. . . . . . . 8 ⊢ (𝑥 ∈ R ↔ 𝑥 ∈ ((P × P) / ~R ))
57	55	eleq2i 2852	. . . . . . . 8 ⊢ (𝑦 ∈ R ↔ 𝑦 ∈ ((P × P) / ~R ))
58	56, 57	anbi12i 640	. . . . . . 7 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ↔ (𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )))
59	58	anbi1i 636	. . . . . 6 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) ↔ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
60	59	oprabbii 7487	. . . . 5 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))}
61	54, 60	eqtri 2783	. . . 4 ⊢ +R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))}
62	52, 53, 61	ovig 7566	. . 3 ⊢ (([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ∈ V) → (∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ))
63	43, 62	mp3an3 1479	. 2 ⊢ (([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )) → (∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ))
64	8, 41, 63	sylc 66	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   × cxp 5648  (class class class)co 7420  {coprab 7421  [cec 8718   / cqs 8719  Pcnp 10947   +_P cpp 10949   ~_R cer 10952  Rcnr 10953   +_R cplr 10957

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-oadd 8483  df-omul 8484  df-er 8720  df-ec 8722  df-qs 8726  df-ni 10960  df-pli 10961  df-mi 10962  df-lti 10963  df-plpq 10996  df-mpq 10997  df-ltpq 10998  df-enq 10999  df-nq 11000  df-erq 11001  df-plq 11002  df-mq 11003  df-1nq 11004  df-rq 11005  df-ltnq 11006  df-np 11069  df-plp 11071  df-ltp 11073  df-enr 11143  df-nr 11144  df-plr 11145

This theorem is used by:  addclsr  11171  addcomsr  11175  addasssr  11176  distrsr  11179  m1p1sr  11180  0idsr  11185  ltasr  11188