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Theorem addsrpr 7743
Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
addsrpr (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨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.
StepHypRef Expression
1 opelxpi 4658 . . . 4 ((𝐴P𝐵P) → ⟨𝐴, 𝐵⟩ ∈ (P × P))
2 enrex 7735 . . . . 5 ~R ∈ V
32ecelqsi 6588 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (P × P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
41, 3syl 14 . . 3 ((𝐴P𝐵P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
5 opelxpi 4658 . . . 4 ((𝐶P𝐷P) → ⟨𝐶, 𝐷⟩ ∈ (P × P))
62ecelqsi 6588 . . . 4 (⟨𝐶, 𝐷⟩ ∈ (P × P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
75, 6syl 14 . . 3 ((𝐶P𝐷P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
84, 7anim12i 338 . 2 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )))
9 eqid 2177 . . . 4 [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R
10 eqid 2177 . . . 4 [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R
119, 10pm3.2i 272 . . 3 ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
12 eqid 2177 . . 3 [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R
13 opeq12 3780 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
1413eceq1d 6570 . . . . . . . 8 ((𝑤 = 𝐴𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝐴, 𝐵⟩] ~R )
1514eqeq2d 2189 . . . . . . 7 ((𝑤 = 𝐴𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ))
1615anbi1d 465 . . . . . 6 ((𝑤 = 𝐴𝑣 = 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 109 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → 𝑤 = 𝐴)
1817oveq1d 5889 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑤 +P 𝐶) = (𝐴 +P 𝐶))
19 simpr 110 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2019oveq1d 5889 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 +P 𝐷) = (𝐵 +P 𝐷))
2118, 20opeq12d 3786 . . . . . . . 8 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩ = ⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩)
2221eceq1d 6570 . . . . . . 7 ((𝑤 = 𝐴𝑣 = 𝐵) → [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
2322eqeq2d 2189 . . . . . 6 ((𝑤 = 𝐴𝑣 = 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 ))
2416, 23anbi12d 473 . . . . 5 ((𝑤 = 𝐴𝑣 = 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 )))
2524spc2egv 2827 . . . 4 ((𝐴P𝐵P) → ((([⟨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 ) → ∃𝑤𝑣(([⟨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 3780 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
2726eceq1d 6570 . . . . . . . . 9 ((𝑢 = 𝐶𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
2827eqeq2d 2189 . . . . . . . 8 ((𝑢 = 𝐶𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
2928anbi2d 464 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 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 109 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → 𝑢 = 𝐶)
3130oveq2d 5890 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑤 +P 𝑢) = (𝑤 +P 𝐶))
32 simpr 110 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → 𝑡 = 𝐷)
3332oveq2d 5890 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 +P 𝑡) = (𝑣 +P 𝐷))
3431, 33opeq12d 3786 . . . . . . . . 9 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩ = ⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩)
3534eceq1d 6570 . . . . . . . 8 ((𝑢 = 𝐶𝑡 = 𝐷) → [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R = [⟨(𝑤 +P 𝐶), (𝑣 +P 𝐷)⟩] ~R )
3635eqeq2d 2189 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 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 ))
3729, 36anbi12d 473 . . . . . 6 ((𝑢 = 𝐶𝑡 = 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 )))
3837spc2egv 2827 . . . . 5 ((𝐶P𝐷P) → ((([⟨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 ) → ∃𝑢𝑡(([⟨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 )))
39382eximdv 1882 . . . 4 ((𝐶P𝐷P) → (∃𝑤𝑣(([⟨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 ) → ∃𝑤𝑣𝑢𝑡(([⟨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 )))
4025, 39sylan9 409 . . 3 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ((([⟨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 ) → ∃𝑤𝑣𝑢𝑡(([⟨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 )))
4111, 12, 40mp2ani 432 . 2 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ∃𝑤𝑣𝑢𝑡(([⟨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 6538 . . . 4 ( ~R ∈ V → [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ∈ V)
432, 42ax-mp 5 . . 3 [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ∈ V
44 simp1 997 . . . . . . . 8 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → 𝑥 = [⟨𝐴, 𝐵⟩] ~R )
4544eqeq1d 2186 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (𝑥 = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ))
46 simp2 998 . . . . . . . 8 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → 𝑦 = [⟨𝐶, 𝐷⟩] ~R )
4746eqeq1d 2186 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (𝑦 = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ))
4845, 47anbi12d 473 . . . . . 6 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ↔ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R )))
49 simp3 999 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → 𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
5049eqeq1d 2186 . . . . . 6 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(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 ))
5148, 50anbi12d 473 . . . . 5 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨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 )))
52514exbidv 1870 . . . 4 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ) → (∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ ∃𝑤𝑣𝑢𝑡(([⟨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 7741 . . . 4 ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
54 df-plr 7726 . . . . 5 +R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))}
55 df-nr 7725 . . . . . . . . 9 R = ((P × P) / ~R )
5655eleq2i 2244 . . . . . . . 8 (𝑥R𝑥 ∈ ((P × P) / ~R ))
5755eleq2i 2244 . . . . . . . 8 (𝑦R𝑦 ∈ ((P × P) / ~R ))
5856, 57anbi12i 460 . . . . . . 7 ((𝑥R𝑦R) ↔ (𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )))
5958anbi1i 458 . . . . . 6 (((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) ↔ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
6059oprabbii 5929 . . . . 5 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))}
6154, 60eqtri 2198 . . . 4 +R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))}
6252, 53, 61ovig 5995 . . 3 (([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R ∈ V) → (∃𝑤𝑣𝑢𝑡(([⟨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 ))
6343, 62mp3an3 1326 . 2 (([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )) → (∃𝑤𝑣𝑢𝑡(([⟨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 ))
648, 41, 63sylc 62 1 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737  cop 3595   × cxp 4624  (class class class)co 5874  {coprab 5875  [cec 6532   / cqs 6533  Pcnp 7289   +P cpp 7291   ~R cer 7294  Rcnr 7295   +R cplr 7299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iplp 7466  df-enr 7724  df-nr 7725  df-plr 7726
This theorem is referenced by:  addclsr  7751  addcomsrg  7753  addasssrg  7754  distrsrg  7757  m1p1sr  7758  0idsr  7765  ltasrg  7768  prsradd  7784  pitonnlem2  7845
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