Theorem mulsrpr 11036

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	⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨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.

Step	Hyp	Ref	Expression
1		opelxpi 5678	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ⟨𝐴, 𝐵⟩ ∈ (P × P))
2		enrex 11027	. . . . 5 ⊢ ~R ∈ V
3	2	ecelqsi 8746	. . . 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 5678	. . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → ⟨𝐶, 𝐷⟩ ∈ (P × P))
6	2	ecelqsi 8746	. . . 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 613	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )))
9		eqid 2730	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R
10		eqid 2730	. . . 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 2730	. . 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 4842	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14	13	eceq1d 8714	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝐴, 𝐵⟩] ~R )
15	14	eqeq2d 2741	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ))
16	15	anbi1d 631	. . . . . 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	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑤 = 𝐴)
18	17	oveq1d 7405	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 ·P 𝐶) = (𝐴 ·P 𝐶))
19		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
20	19	oveq1d 7405	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·P 𝐷) = (𝐵 ·P 𝐷))
21	18, 20	oveq12d 7408	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)) = ((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)))
22	17	oveq1d 7405	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 ·P 𝐷) = (𝐴 ·P 𝐷))
23	19	oveq1d 7405	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·P 𝐶) = (𝐵 ·P 𝐶))
24	22, 23	oveq12d 7408	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶)) = ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶)))
25	21, 24	opeq12d 4848	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 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))⟩)
26	25	eceq1d 8714	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 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 )
27	26	eqeq2d 2741	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 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 ))
28	16, 27	anbi12d 632	. . . . 5 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 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 )))
29	28	spc2egv 3568	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((([⟨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 ) → ∃𝑤∃𝑣(([⟨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 4842	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
31	30	eceq1d 8714	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
32	31	eqeq2d 2741	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
33	32	anbi2d 630	. . . . . . 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 )))
34		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑢 = 𝐶)
35	34	oveq2d 7406	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 ·P 𝑢) = (𝑤 ·P 𝐶))
36		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑡 = 𝐷)
37	36	oveq2d 7406	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·P 𝑡) = (𝑣 ·P 𝐷))
38	35, 37	oveq12d 7408	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)) = ((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)))
39	36	oveq2d 7406	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 ·P 𝑡) = (𝑤 ·P 𝐷))
40	34	oveq2d 7406	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·P 𝑢) = (𝑣 ·P 𝐶))
41	39, 40	oveq12d 7408	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢)) = ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶)))
42	38, 41	opeq12d 4848	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 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))⟩)
43	42	eceq1d 8714	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 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 )
44	43	eqeq2d 2741	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 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 ))
45	33, 44	anbi12d 632	. . . . . 6 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 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 )))
46	45	spc2egv 3568	. . . . 5 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → ((([⟨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 ) → ∃𝑢∃𝑡(([⟨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 )))
47	46	2eximdv 1919	. . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → (∃𝑤∃𝑣(([⟨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 ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨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 )))
48	29, 47	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) +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 ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨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 )))
49	11, 12, 48	mp2ani 698	. 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) +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 8678	. . . 4 ⊢ ( ~R ∈ V → [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ∈ V)
51	2, 50	ax-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⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → 𝑥 = [⟨𝐴, 𝐵⟩] ~R )
53	52	eqeq1d 2732	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((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⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → 𝑦 = [⟨𝐶, 𝐷⟩] ~R )
55	54	eqeq1d 2732	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((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 ))
56	53, 55	anbi12d 632	. . . . . 6 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ↔ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R )))
57		simp3 1138	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((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 )
58	57	eqeq1d 2732	. . . . . 6 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((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 ))
59	56, 58	anbi12d 632	. . . . 5 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨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 )))
60	59	4exbidv 1926	. . . 4 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~R ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~R ∧ 𝑧 = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R ) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨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 11034	. . . 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) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
62		df-mr 11018	. . . . 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) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))}
63		df-nr 11016	. . . . . . . . 9 ⊢ R = ((P × P) / ~R )
64	63	eleq2i 2821	. . . . . . . 8 ⊢ (𝑥 ∈ R ↔ 𝑥 ∈ ((P × P) / ~R ))
65	63	eleq2i 2821	. . . . . . . 8 ⊢ (𝑦 ∈ R ↔ 𝑦 ∈ ((P × P) / ~R ))
66	64, 65	anbi12i 628	. . . . . . 7 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ↔ (𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )))
67	66	anbi1i 624	. . . . . 6 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨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 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
68	67	oprabbii 7459	. . . . 5 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨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 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))}
69	62, 68	eqtri 2753	. . . 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) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))}
70	60, 61, 69	ovig 7538	. . 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) → (∃𝑤∃𝑣∃𝑢∃𝑡(([⟨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 ))
71	51, 70	mp3an3 1452	. 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) +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 ))
72	8, 49, 71	sylc 65	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598   × cxp 5639  (class class class)co 7390  {coprab 7391  [cec 8672   / cqs 8673  Pcnp 10819   +_P cpp 10821   ·_P cmp 10822   ~_R cer 10824  Rcnr 10825   ·_R cmr 10830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-er 8674  df-ec 8676  df-qs 8680  df-ni 10832  df-pli 10833  df-mi 10834  df-lti 10835  df-plpq 10868  df-mpq 10869  df-ltpq 10870  df-enq 10871  df-nq 10872  df-erq 10873  df-plq 10874  df-mq 10875  df-1nq 10876  df-rq 10877  df-ltnq 10878  df-np 10941  df-plp 10943  df-mp 10944  df-ltp 10945  df-enr 11015  df-nr 11016  df-mr 11018

This theorem is referenced by:  mulclsr  11044  mulcomsr  11049  mulasssr  11050  distrsr  11051  m1m1sr  11053  1idsr  11058  00sr  11059  recexsrlem  11063  mulgt0sr  11065