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Theorem mulsrmo 11068

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 11057	. . . . . . . . . . . . . . . 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 11066	. . . . . . . . . . . . . . . 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 11065	. . . . . . . . . . . . . . . . 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 8753	. . . . . . . . . . . . . 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 720	. . . . . . . . . . . . 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 722	. . . . . . . . . . . 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 777	. . . . . . . . . . . 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 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 ))) → 𝑞 = [⟨((𝑠 ·_P 𝑔) +_P (𝑓 ·_P ℎ)), ((𝑠 ·_P ℎ) +_P (𝑓 ·_P 𝑔))⟩] ~_R )
12	9, 10, 11	3eqtr4d 2776	. . . . . . . . . . 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 1929	. . . . . . . . 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 1929	. . . . . . . 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 1929	. . . . . 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 1929	. . . . 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 1923	. . 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 4870	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
22	21	eceq1d 8741	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~_R = [⟨𝑠, 𝑓⟩] ~_R )
23	22	eqeq2d 2737	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~_R ↔ 𝐴 = [⟨𝑠, 𝑓⟩] ~_R ))
24	23	anbi1d 629	. . . . . . . 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 7419	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 ·_P 𝑢) = (𝑠 ·_P 𝑢))
27		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑣 = 𝑓)
28	27	oveq1d 7419	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·_P 𝑡) = (𝑓 ·_P 𝑡))
29	26, 28	oveq12d 7422	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝑤 ·_P 𝑢) +_P (𝑣 ·_P 𝑡)) = ((𝑠 ·_P 𝑢) +_P (𝑓 ·_P 𝑡)))
30	25	oveq1d 7419	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 ·_P 𝑡) = (𝑠 ·_P 𝑡))
31	27	oveq1d 7419	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·_P 𝑢) = (𝑓 ·_P 𝑢))
32	30, 31	oveq12d 7422	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝑤 ·_P 𝑡) +_P (𝑣 ·_P 𝑢)) = ((𝑠 ·_P 𝑡) +_P (𝑓 ·_P 𝑢)))
33	29, 32	opeq12d 4876	. . . . . . . . . 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 8741	. . . . . . . . 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 2737	. . . . . . . 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 630	. . . . . . 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 4870	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ℎ⟩)
38	37	eceq1d 8741	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨𝑢, 𝑡⟩] ~_R = [⟨𝑔, ℎ⟩] ~_R )
39	38	eqeq2d 2737	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~_R ↔ 𝐵 = [⟨𝑔, ℎ⟩] ~_R ))
40	39	anbi2d 628	. . . . . . . 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 7420	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 ·_P 𝑢) = (𝑠 ·_P 𝑔))
43		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑡 = ℎ)
44	43	oveq2d 7420	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·_P 𝑡) = (𝑓 ·_P ℎ))
45	42, 44	oveq12d 7422	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝑠 ·_P 𝑢) +_P (𝑓 ·_P 𝑡)) = ((𝑠 ·_P 𝑔) +_P (𝑓 ·_P ℎ)))
46	43	oveq2d 7420	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 ·_P 𝑡) = (𝑠 ·_P ℎ))
47	41	oveq2d 7420	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·_P 𝑢) = (𝑓 ·_P 𝑔))
48	46, 47	oveq12d 7422	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝑠 ·_P 𝑡) +_P (𝑓 ·_P 𝑢)) = ((𝑠 ·_P ℎ) +_P (𝑓 ·_P 𝑔)))
49	45, 48	opeq12d 4876	. . . . . . . . . 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 8741	. . . . . . . . 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 2737	. . . . . . . 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 630	. . . . . . 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 2037	. . . . . 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 622	. . . . 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 1814	. . 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 233	. 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 2730	. . . . 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 628	. . . 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 1921	. . 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 2554	. 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 233	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 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃*wmo 2526 ⟨cop 4629 class class class wbr 5141 × cxp 5667 (class class class)co 7404 Er wer 8699 [cec 8700 / cqs 8701 Pcnp 10853 +_P cpp 10855 ·_P cmp 10856 ~_R cer 10858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-omul 8469 df-er 8702 df-ec 8704 df-qs 8708 df-ni 10866 df-pli 10867 df-mi 10868 df-lti 10869 df-plpq 10902 df-mpq 10903 df-ltpq 10904 df-enq 10905 df-nq 10906 df-erq 10907 df-plq 10908 df-mq 10909 df-1nq 10910 df-rq 10911 df-ltnq 10912 df-np 10975 df-plp 10977 df-mp 10978 df-ltp 10979 df-enr 11049

This theorem is referenced by: mulsrpr 11070

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