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Theorem mulsrmo 10575
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 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨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 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrer 10564 . . . . . . . . . . . . . . . 16 ~R Er (P × P)
21a1i 11 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ~R Er (P × P))
3 prsrlem1 10573 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔))))
4 mulcmpblnr 10572 . . . . . . . . . . . . . . . . 17 ((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) → (((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔)) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩))
54imp 410 . . . . . . . . . . . . . . . 16 (((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔))) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩)
63, 5syl 17 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩)
72, 6erthi 8372 . . . . . . . . . . . . . 14 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
87adantrlr 723 . . . . . . . . . . . . 13 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
98adantrrr 725 . . . . . . . . . . . 12 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +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 )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
10 simprlr 780 . . . . . . . . . . . 12 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )
11 simprrr 782 . . . . . . . . . . . 12 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
129, 10, 113eqtr4d 2783 . . . . . . . . . . 11 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑧 = 𝑞)
1312expr 460 . . . . . . . . . 10 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
1413exlimdvv 1940 . . . . . . . . 9 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (∃𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
1514exlimdvv 1940 . . . . . . . 8 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
1615ex 416 . . . . . . 7 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
1716exlimdvv 1940 . . . . . 6 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
1817exlimdvv 1940 . . . . 5 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
1918impd 414 . . . 4 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
2019alrimivv 1934 . . 3 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
21 opeq12 4764 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
2221eceq1d 8360 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝑠, 𝑓⟩] ~R )
2322eqeq2d 2749 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐴 = [⟨𝑠, 𝑓⟩] ~R ))
2423anbi1d 633 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R )))
25 simpl 486 . . . . . . . . . . . . 13 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑤 = 𝑠)
2625oveq1d 7186 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑤 ·P 𝑢) = (𝑠 ·P 𝑢))
27 simpr 488 . . . . . . . . . . . . 13 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑣 = 𝑓)
2827oveq1d 7186 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑣 ·P 𝑡) = (𝑓 ·P 𝑡))
2926, 28oveq12d 7189 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)) = ((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)))
3025oveq1d 7186 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑤 ·P 𝑡) = (𝑠 ·P 𝑡))
3127oveq1d 7186 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑣 ·P 𝑢) = (𝑓 ·P 𝑢))
3230, 31oveq12d 7189 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢)) = ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢)))
3329, 32opeq12d 4770 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 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))⟩)
3433eceq1d 8360 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 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 )
3534eqeq2d 2749 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ))
3624, 35anbi12d 634 . . . . . . 7 ((𝑤 = 𝑠𝑣 = 𝑓) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R )))
37 opeq12 4764 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ⟩)
3837eceq1d 8360 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝑔, ⟩] ~R )
3938eqeq2d 2749 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))
4039anbi2d 632 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R )))
41 simpl 486 . . . . . . . . . . . . 13 ((𝑢 = 𝑔𝑡 = ) → 𝑢 = 𝑔)
4241oveq2d 7187 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑠 ·P 𝑢) = (𝑠 ·P 𝑔))
43 simpr 488 . . . . . . . . . . . . 13 ((𝑢 = 𝑔𝑡 = ) → 𝑡 = )
4443oveq2d 7187 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑓 ·P 𝑡) = (𝑓 ·P ))
4542, 44oveq12d 7189 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)) = ((𝑠 ·P 𝑔) +P (𝑓 ·P )))
4643oveq2d 7187 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑠 ·P 𝑡) = (𝑠 ·P ))
4741oveq2d 7187 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑓 ·P 𝑢) = (𝑓 ·P 𝑔))
4846, 47oveq12d 7189 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢)) = ((𝑠 ·P ) +P (𝑓 ·P 𝑔)))
4945, 48opeq12d 4770 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → ⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩ = ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩)
5049eceq1d 8360 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
5150eqeq2d 2749 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → (𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))
5240, 51anbi12d 634 . . . . . . 7 ((𝑢 = 𝑔𝑡 = ) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )))
5336, 52cbvex4vw 2053 . . . . . 6 (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))
5453anbi2i 626 . . . . 5 ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) ↔ (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )))
5554imbi1i 353 . . . 4 (((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞) ↔ ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
56552albii 1827 . . 3 (∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞) ↔ ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
5720, 56sylibr 237 . 2 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞))
58 eqeq1 2742 . . . . 5 (𝑧 = 𝑞 → (𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
5958anbi2d 632 . . . 4 (𝑧 = 𝑞 → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
60594exbidv 1932 . . 3 (𝑧 = 𝑞 → (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
6160mo4 2566 . 2 (∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞))
6257, 61sylibr 237 1 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨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 399  wal 1540   = wceq 1542  wex 1786  wcel 2113  ∃*wmo 2538  cop 4523   class class class wbr 5031   × cxp 5524  (class class class)co 7171   Er wer 8318  [cec 8319   / cqs 8320  Pcnp 10360   +P cpp 10362   ·P cmp 10363   ~R cer 10365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-inf2 9178
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-om 7601  df-1st 7715  df-2nd 7716  df-wrecs 7977  df-recs 8038  df-rdg 8076  df-1o 8132  df-oadd 8136  df-omul 8137  df-er 8321  df-ec 8323  df-qs 8327  df-ni 10373  df-pli 10374  df-mi 10375  df-lti 10376  df-plpq 10409  df-mpq 10410  df-ltpq 10411  df-enq 10412  df-nq 10413  df-erq 10414  df-plq 10415  df-mq 10416  df-1nq 10417  df-rq 10418  df-ltnq 10419  df-np 10482  df-plp 10484  df-mp 10485  df-ltp 10486  df-enr 10556
This theorem is referenced by:  mulsrpr  10577
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