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Theorem mulsrpr 11012
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 5670 . . . 4 ((𝐴P𝐵P) → ⟨𝐴, 𝐵⟩ ∈ (P × P))
2 enrex 11003 . . . . 5 ~R ∈ V
32ecelqsi 8712 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (P × P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
41, 3syl 17 . . 3 ((𝐴P𝐵P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
5 opelxpi 5670 . . . 4 ((𝐶P𝐷P) → ⟨𝐶, 𝐷⟩ ∈ (P × P))
62ecelqsi 8712 . . . 4 (⟨𝐶, 𝐷⟩ ∈ (P × P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
75, 6syl 17 . . 3 ((𝐶P𝐷P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
84, 7anim12i 613 . 2 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )))
9 eqid 2736 . . . 4 [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R
10 eqid 2736 . . . 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 2736 . . 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 4832 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
1413eceq1d 8687 . . . . . . . 8 ((𝑤 = 𝐴𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝐴, 𝐵⟩] ~R )
1514eqeq2d 2747 . . . . . . 7 ((𝑤 = 𝐴𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ))
1615anbi1d 630 . . . . . 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 7372 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑤 ·P 𝐶) = (𝐴 ·P 𝐶))
19 simpr 485 . . . . . . . . . . 11 ((𝑤 = 𝐴𝑣 = 𝐵) → 𝑣 = 𝐵)
2019oveq1d 7372 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 ·P 𝐷) = (𝐵 ·P 𝐷))
2118, 20oveq12d 7375 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)) = ((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)))
2217oveq1d 7372 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑤 ·P 𝐷) = (𝐴 ·P 𝐷))
2319oveq1d 7372 . . . . . . . . . 10 ((𝑤 = 𝐴𝑣 = 𝐵) → (𝑣 ·P 𝐶) = (𝐵 ·P 𝐶))
2422, 23oveq12d 7375 . . . . . . . . 9 ((𝑤 = 𝐴𝑣 = 𝐵) → ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶)) = ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶)))
2521, 24opeq12d 4838 . . . . . . . 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 8687 . . . . . . 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 2747 . . . . . 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 631 . . . . 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 3558 . . . 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 4832 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
3130eceq1d 8687 . . . . . . . . 9 ((𝑢 = 𝐶𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
3231eqeq2d 2747 . . . . . . . 8 ((𝑢 = 𝐶𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
3332anbi2d 629 . . . . . . 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 7373 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑤 ·P 𝑢) = (𝑤 ·P 𝐶))
36 simpr 485 . . . . . . . . . . . 12 ((𝑢 = 𝐶𝑡 = 𝐷) → 𝑡 = 𝐷)
3736oveq2d 7373 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 ·P 𝑡) = (𝑣 ·P 𝐷))
3835, 37oveq12d 7375 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)) = ((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)))
3936oveq2d 7373 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑤 ·P 𝑡) = (𝑤 ·P 𝐷))
4034oveq2d 7373 . . . . . . . . . . 11 ((𝑢 = 𝐶𝑡 = 𝐷) → (𝑣 ·P 𝑢) = (𝑣 ·P 𝐶))
4139, 40oveq12d 7375 . . . . . . . . . 10 ((𝑢 = 𝐶𝑡 = 𝐷) → ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢)) = ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶)))
4238, 41opeq12d 4838 . . . . . . . . 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 8687 . . . . . . . 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 2747 . . . . . . 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 631 . . . . . 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 3558 . . . . 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 1922 . . . 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 696 . 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 8652 . . . 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 1136 . . . . . . . 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 2738 . . . . . . 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 1137 . . . . . . . 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 2738 . . . . . . 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 631 . . . . . 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 1138 . . . . . . 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 2738 . . . . . 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 631 . . . . 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 1929 . . . 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 11010 . . . 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 10994 . . . . 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 10992 . . . . . . . . 9 R = ((P × P) / ~R )
6463eleq2i 2829 . . . . . . . 8 (𝑥R𝑥 ∈ ((P × P) / ~R ))
6563eleq2i 2829 . . . . . . . 8 (𝑦R𝑦 ∈ ((P × P) / ~R ))
6664, 65anbi12i 627 . . . . . . 7 ((𝑥R𝑦R) ↔ (𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )))
6766anbi1i 624 . . . . . 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 7424 . . . . 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 2764 . . . 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 7501 . . 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 1450 . 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 1087   = wceq 1541  wex 1781  wcel 2106  Vcvv 3445  cop 4592   × cxp 5631  (class class class)co 7357  {coprab 7358  [cec 8646   / cqs 8647  Pcnp 10795   +P cpp 10797   ·P cmp 10798   ~R cer 10800  Rcnr 10801   ·R cmr 10806
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ec 8650  df-qs 8654  df-ni 10808  df-pli 10809  df-mi 10810  df-lti 10811  df-plpq 10844  df-mpq 10845  df-ltpq 10846  df-enq 10847  df-nq 10848  df-erq 10849  df-plq 10850  df-mq 10851  df-1nq 10852  df-rq 10853  df-ltnq 10854  df-np 10917  df-plp 10919  df-mp 10920  df-ltp 10921  df-enr 10991  df-nr 10992  df-mr 10994
This theorem is referenced by:  mulclsr  11020  mulcomsr  11025  mulasssr  11026  distrsr  11027  m1m1sr  11029  1idsr  11034  00sr  11035  recexsrlem  11039  mulgt0sr  11041
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