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Theorem mulnq0mo 7422
Description: There is at most one result from multiplying nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
Assertion
Ref Expression
mulnq0mo ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ))
Distinct variable groups:   𝑡,𝐴,𝑢,𝑣,𝑤,𝑧   𝑡,𝐵,𝑢,𝑣,𝑤,𝑧

Proof of Theorem mulnq0mo
Dummy variables 𝑓 𝑔 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enq0er 7409 . . . . . . . . . . . . . 14 ~Q0 Er (ω × N)
21a1i 9 . . . . . . . . . . . . 13 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))) → ~Q0 Er (ω × N))
3 nnnq0lem1 7420 . . . . . . . . . . . . . 14 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))) → ((((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)) ∧ ((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N))) ∧ ((𝑤 ·o 𝑓) = (𝑣 ·o 𝑠) ∧ (𝑢 ·o ) = (𝑡 ·o 𝑔))))
4 mulcmpblnq0 7418 . . . . . . . . . . . . . . 15 ((((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)) ∧ ((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N))) → (((𝑤 ·o 𝑓) = (𝑣 ·o 𝑠) ∧ (𝑢 ·o ) = (𝑡 ·o 𝑔)) → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ ~Q0 ⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩))
54imp 124 . . . . . . . . . . . . . 14 (((((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)) ∧ ((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N))) ∧ ((𝑤 ·o 𝑓) = (𝑣 ·o 𝑠) ∧ (𝑢 ·o ) = (𝑡 ·o 𝑔))) → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ ~Q0 ⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩)
63, 5syl 14 . . . . . . . . . . . . 13 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))) → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ ~Q0 ⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩)
72, 6erthi 6571 . . . . . . . . . . . 12 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))) → [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )
8 simprlr 538 . . . . . . . . . . . 12 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))) → 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )
9 simprrr 540 . . . . . . . . . . . 12 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))) → 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )
107, 8, 93eqtr4d 2218 . . . . . . . . . . 11 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))) → 𝑧 = 𝑞)
1110expr 375 . . . . . . . . . 10 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ) → 𝑧 = 𝑞))
1211exlimdvv 1895 . . . . . . . . 9 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )) → (∃𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ) → 𝑧 = 𝑞))
1312exlimdvv 1895 . . . . . . . 8 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ) → 𝑧 = 𝑞))
1413ex 115 . . . . . . 7 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ) → 𝑧 = 𝑞)))
1514exlimdvv 1895 . . . . . 6 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (∃𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ) → 𝑧 = 𝑞)))
1615exlimdvv 1895 . . . . 5 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ) → 𝑧 = 𝑞)))
1716impd 254 . . . 4 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )) → 𝑧 = 𝑞))
1817alrimivv 1873 . . 3 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )) → 𝑧 = 𝑞))
19 opeq12 3776 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
2019eceq1d 6561 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝑠, 𝑓⟩] ~Q0 )
2120eqeq2d 2187 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ))
2221anbi1d 465 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 )))
23 simpl 109 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑤 = 𝑠)
2423oveq1d 5880 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑤 ·o 𝑢) = (𝑠 ·o 𝑢))
25 simpr 110 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑣 = 𝑓)
2625oveq1d 5880 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑣 ·o 𝑡) = (𝑓 ·o 𝑡))
2724, 26opeq12d 3782 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ = ⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩)
2827eceq1d 6561 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 = [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0 )
2928eqeq2d 2187 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0𝑞 = [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0 ))
3022, 29anbi12d 473 . . . . . . 7 ((𝑤 = 𝑠𝑣 = 𝑓) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0 )))
31 opeq12 3776 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ⟩)
3231eceq1d 6561 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝑔, ⟩] ~Q0 )
3332eqeq2d 2187 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ))
3433anbi2d 464 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 )))
35 simpl 109 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → 𝑢 = 𝑔)
3635oveq2d 5881 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → (𝑠 ·o 𝑢) = (𝑠 ·o 𝑔))
37 simpr 110 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → 𝑡 = )
3837oveq2d 5881 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → (𝑓 ·o 𝑡) = (𝑓 ·o ))
3936, 38opeq12d 3782 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → ⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩ = ⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩)
4039eceq1d 6561 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )
4140eqeq2d 2187 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → (𝑞 = [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))
4234, 41anbi12d 473 . . . . . . 7 ((𝑢 = 𝑔𝑡 = ) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )))
4330, 42cbvex4v 1928 . . . . . 6 (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ↔ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))
4443anbi2i 457 . . . . 5 ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )) ↔ (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )))
4544imbi1i 238 . . . 4 (((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞) ↔ ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )) → 𝑧 = 𝑞))
46452albii 1469 . . 3 (∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞) ↔ ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )) → 𝑧 = 𝑞))
4718, 46sylibr 134 . 2 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞))
48 eqeq1 2182 . . . . 5 (𝑧 = 𝑞 → (𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ))
4948anbi2d 464 . . . 4 (𝑧 = 𝑞 → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )))
50494exbidv 1868 . . 3 (𝑧 = 𝑞 → (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ↔ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )))
5150mo4 2085 . 2 (∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ↔ ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞))
5247, 51sylibr 134 1 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  wex 1490  ∃*wmo 2025  wcel 2146  cop 3592   class class class wbr 3998  ωcom 4583   × cxp 4618  (class class class)co 5865   ·o comu 6405   Er wer 6522  [cec 6523   / cqs 6524  Ncnpi 7246   ~Q0 ceq0 7260
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-mi 7280  df-enq0 7398
This theorem is referenced by:  mulnnnq0  7424
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