Theorem addsrpr 10989

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 5661	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ⟨𝐴, 𝐵⟩ ∈ (P × P))
2		enrex 10981	. . . . 5 ⊢ ~R ∈ V
3	2	ecelqsi 8709	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (P × P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
4	1, 3	syl 17	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
5		opelxpi 5661	. . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → ⟨𝐶, 𝐷⟩ ∈ (P × P))
6	2	ecelqsi 8709	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ ∈ (P × P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
7	5, 6	syl 17	. . 3 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
8	4, 7	anim12i 614	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )))
9		eqid 2737	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R
10		eqid 2737	. . . 4 ⊢ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R
11	9, 10	pm3.2i 470	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
12		eqid 2737	. . 3 ⊢ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R
13		opeq12 4819	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14	13	eceq1d 8677	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝐴, 𝐵⟩] ~R )
15	14	eqeq2d 2748	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ))
16	15	anbi1d 632	. . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑤 = 𝐴)
18	17	oveq1d 7375	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 +P 𝐶) = (𝐴 +P 𝐶))
19		simpr 484	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
20	19	oveq1d 7375	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 +P 𝐷) = (𝐵 +P 𝐷))
21	18, 20	opeq12d 4825	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩ = ⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩)
22	21	eceq1d 8677	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
23	22	eqeq2d 2748	. . . . . 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 633	. . . . 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 3542	. . . 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 4819	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
27	26	eceq1d 8677	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
28	27	eqeq2d 2748	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
29	28	anbi2d 631	. . . . . . 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 482	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑢 = 𝐶)
31	30	oveq2d 7376	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 +P 𝑢) = (𝑤 +P 𝐶))
32		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑡 = 𝐷)
33	32	oveq2d 7376	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 +P 𝑡) = (𝑣 +P 𝐷))
34	31, 33	opeq12d 4825	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩ = ⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩)
35	34	eceq1d 8677	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R )
36	35	eqeq2d 2748	. . . . . . 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 633	. . . . . 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 3542	. . . . 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 1921	. . . 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 507	. . 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 699	. 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 8640	. . . 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 1137	. . . . . . . 8 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → 𝑥 = [⟨𝐴, 𝐵⟩] ~R )
45	44	eqeq1d 2739	. . . . . . 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 1138	. . . . . . . 8 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → 𝑦 = [⟨𝐶, 𝐷⟩] ~R )
47	46	eqeq1d 2739	. . . . . . 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 633	. . . . . 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 1139	. . . . . . 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 2739	. . . . . 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 633	. . . . 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 1928	. . . 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 10987	. . . 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 10971	. . . . 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 10970	. . . . . . . . 9 ⊢ R = ((P × P) / ~R )
56	55	eleq2i 2829	. . . . . . . 8 ⊢ (𝑥 ∈ R ↔ 𝑥 ∈ ((P × P) / ~R ))
57	55	eleq2i 2829	. . . . . . . 8 ⊢ (𝑦 ∈ R ↔ 𝑦 ∈ ((P × P) / ~R ))
58	56, 57	anbi12i 629	. . . . . . 7 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ↔ (𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )))
59	58	anbi1i 625	. . . . . 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 7427	. . . . 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 2760	. . . 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 7506	. . 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 1453	. 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 65	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   × cxp 5622  (class class class)co 7360  {coprab 7361  [cec 8634   / cqs 8635  Pcnp 10773   +_P cpp 10775   ~_R cer 10778  Rcnr 10779   +_R cplr 10783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-oadd 8402  df-omul 8403  df-er 8636  df-ec 8638  df-qs 8642  df-ni 10786  df-pli 10787  df-mi 10788  df-lti 10789  df-plpq 10822  df-mpq 10823  df-ltpq 10824  df-enq 10825  df-nq 10826  df-erq 10827  df-plq 10828  df-mq 10829  df-1nq 10830  df-rq 10831  df-ltnq 10832  df-np 10895  df-plp 10897  df-ltp 10899  df-enr 10969  df-nr 10970  df-plr 10971

This theorem is referenced by:  addclsr  10997  addcomsr  11001  addasssr  11002  distrsr  11005  m1p1sr  11006  0idsr  11011  ltasr  11014