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Theorem mulsrpr 10351
Description: Multiplication 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
mulsrpr (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )

Proof of Theorem mulsrpr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 5487 . . . 4 ((𝐴P𝐵P) → ⟨𝐴, 𝐵⟩ ∈ (P × P))
2 enrex 10342 . . . . 5 ~R ∈ V
32ecelqsi 8210 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (P × P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
41, 3syl 17 . . 3 ((𝐴P𝐵P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
5 opelxpi 5487 . . . 4 ((𝐶P𝐷P) → ⟨𝐶, 𝐷⟩ ∈ (P × P))
62ecelqsi 8210 . . . 4 (⟨𝐶, 𝐷⟩ ∈ (P × P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
75, 6syl 17 . . 3 ((𝐶P𝐷P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
84, 7anim12i 612 . 2 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )))
9 eqid 2797 . . . 4 [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R
10 eqid 2797 . . . 4 [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R
119, 10pm3.2i 471 . . 3 ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
12 eqid 2797 . . 3 [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R
13 opeq12 4718 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
1413eceq1d 8185 . . . . . . . 8 ((𝑤 = 𝐴𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝐴, 𝐵⟩] ~R )
1514eqeq2d 2807 . . . . . . 7 ((𝑤 = 𝐴𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ))
1615anbi1d 629 . . . . . 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 483 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → 𝑤 = 𝐴)
1817oveq1d 7038 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑤 ·P 𝐶) = (𝐴 ·P 𝐶))
19 simpr 485 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2019oveq1d 7038 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 ·P 𝐷) = (𝐵 ·P 𝐷))
2118, 20oveq12d 7041 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)) = ((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)))
2217oveq1d 7038 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑤 ·P 𝐷) = (𝐴 ·P 𝐷))
2319oveq1d 7038 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 ·P 𝐶) = (𝐵 ·P 𝐶))
2422, 23oveq12d 7041 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶)) = ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶)))
2521, 24opeq12d 4724 . . . . . . . 8 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩ = ⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩)
2625eceq1d 8185 . . . . . . 7 ((𝑤 = 𝐴𝑣 = 𝐵) → [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )
2726eqeq2d 2807 . . . . . 6 ((𝑤 = 𝐴𝑣 = 𝐵) → ([⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R ↔ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ))
2816, 27anbi12d 630 . . . . 5 ((𝑤 = 𝐴𝑣 = 𝐵) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R ) ↔ (([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )))
2928spc2egv 3544 . . . 4 ((𝐴P𝐵P) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → ∃𝑤𝑣(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R )))
30 opeq12 4718 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
3130eceq1d 8185 . . . . . . . . 9 ((𝑢 = 𝐶𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
3231eqeq2d 2807 . . . . . . . 8 ((𝑢 = 𝐶𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
3332anbi2d 628 . . . . . . 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 )))
34 simpl 483 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → 𝑢 = 𝐶)
3534oveq2d 7039 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑤 ·P 𝑢) = (𝑤 ·P 𝐶))
36 simpr 485 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → 𝑡 = 𝐷)
3736oveq2d 7039 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 ·P 𝑡) = (𝑣 ·P 𝐷))
3835, 37oveq12d 7041 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)) = ((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)))
3936oveq2d 7039 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑤 ·P 𝑡) = (𝑤 ·P 𝐷))
4034oveq2d 7039 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 ·P 𝑢) = (𝑣 ·P 𝐶))
4139, 40oveq12d 7041 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢)) = ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶)))
4238, 41opeq12d 4724 . . . . . . . . 9 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ = ⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩)
4342eceq1d 8185 . . . . . . . 8 ((𝑢 = 𝐶𝑡 = 𝐷) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R )
4443eqeq2d 2807 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 𝐷) → ([⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ↔ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R ))
4533, 44anbi12d 630 . . . . . 6 ((𝑢 = 𝐶𝑡 = 𝐷) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ (([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R )))
4645spc2egv 3544 . . . . 5 ((𝐶P𝐷P) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R ) → ∃𝑢𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
47462eximdv 1901 . . . 4 ((𝐶P𝐷P) → (∃𝑤𝑣(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)), ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶))⟩] ~R ) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
4829, 47sylan9 508 . . 3 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ((([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
4911, 12, 48mp2ani 694 . 2 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
50 ecexg 8150 . . . 4 ( ~R ∈ V → [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ∈ V)
512, 50ax-mp 5 . . 3 [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ∈ V
52 simp1 1129 . . . . . . . 8 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → 𝑥 = [⟨𝐴, 𝐵⟩] ~R )
5352eqeq1d 2799 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → (𝑥 = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ))
54 simp2 1130 . . . . . . . 8 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → 𝑦 = [⟨𝐶, 𝐷⟩] ~R )
5554eqeq1d 2799 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → (𝑦 = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ))
5653, 55anbi12d 630 . . . . . 6 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ↔ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R )))
57 simp3 1131 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → 𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )
5857eqeq1d 2799 . . . . . 6 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → (𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ↔ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
5956, 58anbi12d 630 . . . . 5 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ (([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
60594exbidv 1908 . . . 4 ((𝑥 = [⟨𝐴, 𝐵⟩] ~R𝑦 = [⟨𝐶, 𝐷⟩] ~R𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → (∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
61 mulsrmo 10349 . . . 4 ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
62 df-mr 10333 . . . . 5 ·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))}
63 df-nr 10331 . . . . . . . . 9 R = ((P × P) / ~R )
6463eleq2i 2876 . . . . . . . 8 (𝑥R𝑥 ∈ ((P × P) / ~R ))
6563eleq2i 2876 . . . . . . . 8 (𝑦R𝑦 ∈ ((P × P) / ~R ))
6664, 65anbi12i 626 . . . . . . 7 ((𝑥R𝑦R) ↔ (𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )))
6766anbi1i 623 . . . . . 6 (((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) ↔ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
6867oprabbii 7087 . . . . 5 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))}
6962, 68eqtri 2821 . . . 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) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))}
7060, 61, 69ovig 7159 . . 3 (([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ∈ V) → (∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ) ∧ [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ))
7151, 70mp3an3 1442 . 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) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ))
728, 49, 71sylc 65 1 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1525  wex 1765  wcel 2083  Vcvv 3440  cop 4484   × cxp 5448  (class class class)co 7023  {coprab 7024  [cec 8144   / cqs 8145  Pcnp 10134   +P cpp 10136   ·P cmp 10137   ~R cer 10139  Rcnr 10140   ·R cmr 10145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-omul 7965  df-er 8146  df-ec 8148  df-qs 8152  df-ni 10147  df-pli 10148  df-mi 10149  df-lti 10150  df-plpq 10183  df-mpq 10184  df-ltpq 10185  df-enq 10186  df-nq 10187  df-erq 10188  df-plq 10189  df-mq 10190  df-1nq 10191  df-rq 10192  df-ltnq 10193  df-np 10256  df-plp 10258  df-mp 10259  df-ltp 10260  df-enr 10330  df-nr 10331  df-mr 10333
This theorem is referenced by:  mulclsr  10359  mulcomsr  10364  mulasssr  10365  distrsr  10366  m1m1sr  10368  1idsr  10373  00sr  10374  recexsrlem  10378  mulgt0sr  10380
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