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Theorem mulnq0mo 7279
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 7266 . . . . . . . . . . . . . 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 7277 . . . . . . . . . . . . . 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 7275 . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . 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 6482 . . . . . . . . . . . 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 528 . . . . . . . . . . . 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 530 . . . . . . . . . . . 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 2183 . . . . . . . . . . 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 373 . . . . . . . . . 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 1870 . . . . . . . . 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 1870 . . . . . . . 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 114 . . . . . . 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 1870 . . . . . 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 1870 . . . . 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 252 . . . 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 1848 . . 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 3714 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
2019eceq1d 6472 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝑠, 𝑓⟩] ~Q0 )
2120eqeq2d 2152 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ))
2221anbi1d 461 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 )))
23 simpl 108 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑤 = 𝑠)
2423oveq1d 5796 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑤 ·o 𝑢) = (𝑠 ·o 𝑢))
25 simpr 109 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑣 = 𝑓)
2625oveq1d 5796 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑣 ·o 𝑡) = (𝑓 ·o 𝑡))
2724, 26opeq12d 3720 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ = ⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩)
2827eceq1d 6472 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 = [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0 )
2928eqeq2d 2152 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0𝑞 = [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0 ))
3022, 29anbi12d 465 . . . . . . 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 3714 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ⟩)
3231eceq1d 6472 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝑔, ⟩] ~Q0 )
3332eqeq2d 2152 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ))
3433anbi2d 460 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 )))
35 simpl 108 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → 𝑢 = 𝑔)
3635oveq2d 5797 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → (𝑠 ·o 𝑢) = (𝑠 ·o 𝑔))
37 simpr 109 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → 𝑡 = )
3837oveq2d 5797 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → (𝑓 ·o 𝑡) = (𝑓 ·o ))
3936, 38opeq12d 3720 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → ⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩ = ⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩)
4039eceq1d 6472 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 )
4140eqeq2d 2152 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → (𝑞 = [⟨(𝑠 ·o 𝑢), (𝑓 ·o 𝑡)⟩] ~Q0𝑞 = [⟨(𝑠 ·o 𝑔), (𝑓 ·o )⟩] ~Q0 ))
4234, 41anbi12d 465 . . . . . . 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 1903 . . . . . 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 453 . . . . 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 237 . . . 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 1448 . . 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 133 . 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 2147 . . . . 5 (𝑧 = 𝑞 → (𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0𝑞 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ))
4948anbi2d 460 . . . 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 1843 . . 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 2061 . 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 133 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 103  wal 1330   = wceq 1332  wex 1469  wcel 1481  ∃*wmo 2001  cop 3534   class class class wbr 3936  ωcom 4511   × cxp 4544  (class class class)co 5781   ·o comu 6318   Er wer 6433  [cec 6434   / cqs 6435  Ncnpi 7103   ~Q0 ceq0 7117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-mi 7137  df-enq0 7255
This theorem is referenced by:  mulnnnq0  7281
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