Theorem mulsrpr 7966

Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

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 4757	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ⟨𝐴, 𝐵⟩ ∈ (P × P))
2		enrex 7957	. . . . 5 ⊢ ~R ∈ V
3	2	ecelqsi 6758	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (P × P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
4	1, 3	syl 14	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
5		opelxpi 4757	. . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → ⟨𝐶, 𝐷⟩ ∈ (P × P))
6	2	ecelqsi 6758	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ ∈ (P × P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
7	5, 6	syl 14	. . 3 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R ))
8	4, 7	anim12i 338	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ) ∧ [⟨𝐶, 𝐷⟩] ~R ∈ ((P × P) / ~R )))
9		eqid 2231	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R
10		eqid 2231	. . . 4 ⊢ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R
11	9, 10	pm3.2i 272	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ∧ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
12		eqid 2231	. . 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 3864	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14	13	eceq1d 6738	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝐴, 𝐵⟩] ~R )
15	14	eqeq2d 2243	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝑤, 𝑣⟩] ~R ↔ [⟨𝐴, 𝐵⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ))
16	15	anbi1d 465	. . . . . 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 109	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑤 = 𝐴)
18	17	oveq1d 6033	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 ·P 𝐶) = (𝐴 ·P 𝐶))
19		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
20	19	oveq1d 6033	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·P 𝐷) = (𝐵 ·P 𝐷))
21	18, 20	oveq12d 6036	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)) = ((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)))
22	17	oveq1d 6033	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 ·P 𝐷) = (𝐴 ·P 𝐷))
23	19	oveq1d 6033	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·P 𝐶) = (𝐵 ·P 𝐶))
24	22, 23	oveq12d 6036	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶)) = ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶)))
25	21, 24	opeq12d 3870	. . . . . . . 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 6738	. . . . . . 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 2243	. . . . . 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 473	. . . . 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 2896	. . . 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 3864	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
31	30	eceq1d 6738	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
32	31	eqeq2d 2243	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨𝑢, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
33	32	anbi2d 464	. . . . . . 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 109	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑢 = 𝐶)
35	34	oveq2d 6034	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 ·P 𝑢) = (𝑤 ·P 𝐶))
36		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑡 = 𝐷)
37	36	oveq2d 6034	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·P 𝑡) = (𝑣 ·P 𝐷))
38	35, 37	oveq12d 6036	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)) = ((𝑤 ·P 𝐶) +P (𝑣 ·P 𝐷)))
39	36	oveq2d 6034	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 ·P 𝑡) = (𝑤 ·P 𝐷))
40	34	oveq2d 6034	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·P 𝑢) = (𝑣 ·P 𝐶))
41	39, 40	oveq12d 6036	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢)) = ((𝑤 ·P 𝐷) +P (𝑣 ·P 𝐶)))
42	38, 41	opeq12d 3870	. . . . . . . . 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 6738	. . . . . . . 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 2243	. . . . . . 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 473	. . . . . 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 2896	. . . . 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 1930	. . . 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 409	. . 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 432	. 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 6706	. . . 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 1023	. . . . . . . 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 2240	. . . . . . 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 1024	. . . . . . . 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 2240	. . . . . . 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 473	. . . . . 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 1025	. . . . . . 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 2240	. . . . . 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 473	. . . . 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 1918	. . . 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 7964	. . . 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 7949	. . . . 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 7947	. . . . . . . . 9 ⊢ R = ((P × P) / ~R )
64	63	eleq2i 2298	. . . . . . . 8 ⊢ (𝑥 ∈ R ↔ 𝑥 ∈ ((P × P) / ~R ))
65	63	eleq2i 2298	. . . . . . . 8 ⊢ (𝑦 ∈ R ↔ 𝑦 ∈ ((P × P) / ~R ))
66	64, 65	anbi12i 460	. . . . . . 7 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ↔ (𝑥 ∈ ((P × P) / ~R ) ∧ 𝑦 ∈ ((P × P) / ~R )))
67	66	anbi1i 458	. . . . . 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 6076	. . . . 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 2252	. . . 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 6143	. . 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 1362	. 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 62	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 104   ∧ w3a 1004   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802  ⟨cop 3672   × cxp 4723  (class class class)co 6018  {coprab 6019  [cec 6700   / cqs 6701  Pcnp 7511   +_P cpp 7513   ·_P cmp 7514   ~_R cer 7516  Rcnr 7517   ·_R cmr 7522

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-iplp 7688  df-imp 7689  df-enr 7946  df-nr 7947  df-mr 7949

This theorem is referenced by:  mulclsr  7974  mulcomsrg  7977  mulasssrg  7978  distrsrg  7979  m1m1sr  7981  1idsr  7988  00sr  7989  recexgt0sr  7993  mulgt0sr  7998  mulextsr1  8001  recidpirq  8078