Theorem mulsrmo 11115

Description: There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)

Assertion

Ref	Expression
mulsrmo	⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))

Distinct variable groups:   𝑡,𝐴,𝑢,𝑣,𝑤,𝑧   𝑡,𝐵,𝑢,𝑣,𝑤,𝑧

Proof of Theorem mulsrmo

Dummy variables 𝑓 𝑔 ℎ 𝑞 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrer 11104	. . . . . . . . . . . . . . . 16 ⊢ ~R Er (P × P)
2	1	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ~R Er (P × P))
3		prsrlem1 11113	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))))
4		mulcmpblnr 11112	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) → (((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔)) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩))
5	4	imp 406	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩)
6	3, 5	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩)
7	2, 6	erthi 8799	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
8	7	adantrlr 723	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
9	8	adantrrr 725	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
10		simprlr 779	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )
11		simprrr 781	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
12	9, 10, 11	3eqtr4d 2786	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑧 = 𝑞)
13	12	expr 456	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
14	13	exlimdvv 1933	. . . . . . . . 9 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
15	14	exlimdvv 1933	. . . . . . . 8 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
16	15	ex 412	. . . . . . 7 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
17	16	exlimdvv 1933	. . . . . 6 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
18	17	exlimdvv 1933	. . . . 5 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
19	18	impd 410	. . . 4 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
20	19	alrimivv 1927	. . 3 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
21		opeq12 4874	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
22	21	eceq1d 8786	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝑠, 𝑓⟩] ~R )
23	22	eqeq2d 2747	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~R ↔ 𝐴 = [⟨𝑠, 𝑓⟩] ~R ))
24	23	anbi1d 631	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R )))
25		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑤 = 𝑠)
26	25	oveq1d 7447	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 ·P 𝑢) = (𝑠 ·P 𝑢))
27		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑣 = 𝑓)
28	27	oveq1d 7447	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·P 𝑡) = (𝑓 ·P 𝑡))
29	26, 28	oveq12d 7450	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)) = ((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)))
30	25	oveq1d 7447	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 ·P 𝑡) = (𝑠 ·P 𝑡))
31	27	oveq1d 7447	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·P 𝑢) = (𝑓 ·P 𝑢))
32	30, 31	oveq12d 7450	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢)) = ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢)))
33	29, 32	opeq12d 4880	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ = ⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩)
34	33	eceq1d 8786	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R )
35	34	eqeq2d 2747	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ↔ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ))
36	24, 35	anbi12d 632	. . . . . . 7 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R )))
37		opeq12 4874	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ℎ⟩)
38	37	eceq1d 8786	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝑔, ℎ⟩] ~R )
39	38	eqeq2d 2747	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~R ↔ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))
40	39	anbi2d 630	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R )))
41		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑢 = 𝑔)
42	41	oveq2d 7448	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 ·P 𝑢) = (𝑠 ·P 𝑔))
43		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑡 = ℎ)
44	43	oveq2d 7448	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·P 𝑡) = (𝑓 ·P ℎ))
45	42, 44	oveq12d 7450	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)) = ((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)))
46	43	oveq2d 7448	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 ·P 𝑡) = (𝑠 ·P ℎ))
47	41	oveq2d 7448	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·P 𝑢) = (𝑓 ·P 𝑔))
48	46, 47	oveq12d 7450	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢)) = ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔)))
49	45, 48	opeq12d 4880	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩ = ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩)
50	49	eceq1d 8786	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )
51	50	eqeq2d 2747	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ↔ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))
52	40, 51	anbi12d 632	. . . . . . 7 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )))
53	36, 52	cbvex4vw 2040	. . . . . 6 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R ))
54	53	anbi2i 623	. . . . 5 ⊢ ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) ↔ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )))
55	54	imbi1i 349	. . . 4 ⊢ (((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞) ↔ ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
56	55	2albii 1819	. . 3 ⊢ (∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞) ↔ ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P ℎ)), ((𝑠 ·P ℎ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
57	20, 56	sylibr 234	. 2 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞))
58		eqeq1 2740	. . . . 5 ⊢ (𝑧 = 𝑞 → (𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ↔ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
59	58	anbi2d 630	. . . 4 ⊢ (𝑧 = 𝑞 → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
60	59	4exbidv 1925	. . 3 ⊢ (𝑧 = 𝑞 → (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
61	60	mo4 2565	. 2 ⊢ (∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞))
62	57, 61	sylibr 234	1 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃*wmo 2537  ⟨cop 4631   class class class wbr 5142   × cxp 5682  (class class class)co 7432   Er wer 8743  [cec 8744   / cqs 8745  Pcnp 10900   +_P cpp 10902   ·_P cmp 10903   ~_R cer 10905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8746  df-ec 8748  df-qs 8752  df-ni 10913  df-pli 10914  df-mi 10915  df-lti 10916  df-plpq 10949  df-mpq 10950  df-ltpq 10951  df-enq 10952  df-nq 10953  df-erq 10954  df-plq 10955  df-mq 10956  df-1nq 10957  df-rq 10958  df-ltnq 10959  df-np 11022  df-plp 11024  df-mp 11025  df-ltp 11026  df-enr 11096

This theorem is referenced by:  mulsrpr  11117