ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addnq0mo GIF version

Theorem addnq0mo 6602
Description: There is at most one result from adding non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
addnq0mo ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
Distinct variable groups:   𝑡,𝐴,𝑢,𝑣,𝑤,𝑧   𝑡,𝐵,𝑢,𝑣,𝑤,𝑧

Proof of Theorem addnq0mo
Dummy variables 𝑓 𝑔 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enq0er 6590 . . . . . . . . . . . . . 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 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ))) → ~Q0 Er (ω × N))
3 nnnq0lem1 6601 . . . . . . . . . . . . . 14 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·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 addcmpblnq0 6598 . . . . . . . . . . . . . . 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 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩))
54imp 119 . . . . . . . . . . . . . 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 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·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 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ))) → ⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩)
72, 6erthi 6182 . . . . . . . . . . . 12 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ))) → [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )
8 simprlr 498 . . . . . . . . . . . 12 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ))) → 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )
9 simprrr 500 . . . . . . . . . . . 12 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ))) → 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )
107, 8, 93eqtr4d 2098 . . . . . . . . . . 11 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ))) → 𝑧 = 𝑞)
1110expr 361 . . . . . . . . . 10 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ) → 𝑧 = 𝑞))
1211exlimdvv 1793 . . . . . . . . 9 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → (∃𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ) → 𝑧 = 𝑞))
1312exlimdvv 1793 . . . . . . . 8 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ) → 𝑧 = 𝑞))
1413ex 112 . . . . . . 7 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ) → 𝑧 = 𝑞)))
1514exlimdvv 1793 . . . . . 6 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (∃𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ) → 𝑧 = 𝑞)))
1615exlimdvv 1793 . . . . 5 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ) → 𝑧 = 𝑞)))
1716impd 246 . . . 4 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )) → 𝑧 = 𝑞))
1817alrimivv 1771 . . 3 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )) → 𝑧 = 𝑞))
19 opeq12 3578 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
2019eceq1d 6172 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝑠, 𝑓⟩] ~Q0 )
2120eqeq2d 2067 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ))
2221anbi1d 446 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 )))
23 simpl 106 . . . . . . . . . . . . 13 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑤 = 𝑠)
2423oveq1d 5554 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑤 ·𝑜 𝑡) = (𝑠 ·𝑜 𝑡))
25 simpr 107 . . . . . . . . . . . . 13 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑣 = 𝑓)
2625oveq1d 5554 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑣 ·𝑜 𝑢) = (𝑓 ·𝑜 𝑢))
2724, 26oveq12d 5557 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)) = ((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)))
2825oveq1d 5554 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑣 ·𝑜 𝑡) = (𝑓 ·𝑜 𝑡))
2927, 28opeq12d 3584 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ = ⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩)
3029eceq1d 6172 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 )
3130eqeq2d 2067 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 ))
3222, 31anbi12d 450 . . . . . . 7 ((𝑤 = 𝑠𝑣 = 𝑓) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 )))
33 opeq12 3578 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ⟩)
3433eceq1d 6172 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝑔, ⟩] ~Q0 )
3534eqeq2d 2067 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ))
3635anbi2d 445 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 )))
37 simpr 107 . . . . . . . . . . . . 13 ((𝑢 = 𝑔𝑡 = ) → 𝑡 = )
3837oveq2d 5555 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑠 ·𝑜 𝑡) = (𝑠 ·𝑜 ))
39 simpl 106 . . . . . . . . . . . . 13 ((𝑢 = 𝑔𝑡 = ) → 𝑢 = 𝑔)
4039oveq2d 5555 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑓 ·𝑜 𝑢) = (𝑓 ·𝑜 𝑔))
4138, 40oveq12d 5557 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)) = ((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)))
4237oveq2d 5555 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → (𝑓 ·𝑜 𝑡) = (𝑓 ·𝑜 ))
4341, 42opeq12d 3584 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → ⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩ = ⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩)
4443eceq1d 6172 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )
4544eqeq2d 2067 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → (𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ))
4636, 45anbi12d 450 . . . . . . 7 ((𝑢 = 𝑔𝑡 = ) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )))
4732, 46cbvex4v 1821 . . . . . 6 (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 ))
4847anbi2i 438 . . . . 5 ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) ↔ (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )))
4948imbi1i 231 . . . 4 (((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞) ↔ ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )) → 𝑧 = 𝑞))
50492albii 1376 . . 3 (∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞) ↔ ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 )⟩] ~Q0 )) → 𝑧 = 𝑞))
5118, 50sylibr 141 . 2 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞))
52 eqeq1 2062 . . . . 5 (𝑧 = 𝑞 → (𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
5352anbi2d 445 . . . 4 (𝑧 = 𝑞 → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
54534exbidv 1766 . . 3 (𝑧 = 𝑞 → (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
5554mo4 1977 . 2 (∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞))
5651, 55sylibr 141 1 ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257   = wceq 1259  wex 1397  wcel 1409  ∃*wmo 1917  cop 3405   class class class wbr 3791  ωcom 4340   × cxp 4370  (class class class)co 5539   +𝑜 coa 6028   ·𝑜 comu 6029   Er wer 6133  [cec 6134   / cqs 6135  Ncnpi 6427   ~Q0 ceq0 6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-mi 6461  df-enq0 6579
This theorem is referenced by:  addnnnq0  6604
  Copyright terms: Public domain W3C validator