Proofgold Signed Transaction

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition explicit_Nats^{explicit_Nats} := λ x0 x1 . λ x2 : ι → ι . and (and (and (and (x1 ∈ x0) (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0)) (∀ x3 . x3 ∈ x0 ⟶ x2 x3 = x1 ⟶ ∀ x4 : ο . x4)) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4)
Known and5I^and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4
Theorem explicit_Nats_I^{explicit_Nats_I} : ∀ x0 x1 . ∀ x2 : ι → ι . x1 ∈ x0 ⟶ (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0) ⟶ (∀ x3 . x3 ∈ x0 ⟶ x2 x3 = x1 ⟶ ∀ x4 : ο . x4) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ (∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4) ⟶ explicit_Nats x0 x1 x2 (proof)
Known and5E^and5E : ∀ x0 x1 x2 x3 x4 : ο . and (and (and (and x0 x1) x2) x3) x4 ⟶ ∀ x5 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5) ⟶ x5
Theorem explicit_Nats_E^{explicit_Nats_E} : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ο . (explicit_Nats x0 x1 x2 ⟶ x1 ∈ x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x0) ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 : ι → ο . x4 x1 ⟶ (∀ x5 . x4 x5 ⟶ x4 (x2 x5)) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5) ⟶ x3) ⟶ explicit_Nats x0 x1 x2 ⟶ x3 (proof)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem explicit_Nats_ind^{explicit_Nats_ind} : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4 (proof)
Param omega^omega : ι
Param ordsucc^ordsucc : ι → ι
Known e4648.. : 0 ∈ omega
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known neq_ordsucc_0^{neq_ordsucc_0} : ∀ x0 . ordsucc x0 = 0 ⟶ ∀ x1 : ο . x1
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Definition nat_p^nat_p := λ x0 . ∀ x1 : ι → ο . x1 0 ⟶ (∀ x2 . x1 x2 ⟶ x1 (ordsucc x2)) ⟶ x1 x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem explicit_Nats_omega^{explicit_Nats_omega} : explicit_Nats omega 0 ordsucc (proof)
Definition bij^bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Theorem explicit_Nats_transfer^{explicit_Nats_transfer} : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 x4 . ∀ x5 x6 : ι → ι . explicit_Nats x0 x1 x2 ⟶ bij x0 x3 x6 ⟶ x6 x1 = x4 ⟶ (∀ x7 . x7 ∈ x0 ⟶ x6 (x2 x7) = x5 (x6 x7)) ⟶ explicit_Nats x3 x4 x5 (proof)
Definition explicit_Field^{explicit_Field} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . and (and (and (and (and (and (and (and (and (and (and (and (and (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5)) (x1 ∈ x0)) (∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x4 x6 x5)) (x2 ∈ x0)) (x2 = x1 ⟶ ∀ x5 : ο . x5)) (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5)) (∀ x5 . x5 ∈ x0 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x4 x5 x7 = x2) ⟶ x6) ⟶ x6)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7))
Known and7I^and7I : ∀ x0 x1 x2 x3 x4 x5 x6 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6
Theorem explicit_Field_I^{explicit_Field_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5) ⟶ x1 ∈ x0 ⟶ (∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x4 x6 x5) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x4 x5 x7 = x2) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ explicit_Field x0 x1 x2 x3 x4 (proof)
Known and7E^and7E : ∀ x0 x1 x2 x3 x4 x5 x6 : ο . and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6 ⟶ ∀ x7 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ x7) ⟶ x7
Theorem explicit_Field_E^{explicit_Field_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x3 x7 x6) ⟶ x1 ∈ x0 ⟶ (∀ x6 . x6 ∈ x0 ⟶ x3 x1 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x3 x6 x8 = x1) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 = x4 x7 x6) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ (x6 = x1 ⟶ ∀ x7 : ο . x7) ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x4 x6 x8 = x2) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ x5) ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ x5 (proof)
Definition explicit_Field_minus^{explicit_Field_minus} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 . prim0 (λ x6 . and (x6 ∈ x0) (x3 x5 x6 = x1))
Known Eps_i_ex^Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0)
Theorem explicit_Field_minus_prop^{explicit_Field_minus_prop} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ and (explicit_Field_minus x0 x1 x2 x3 x4 x5 ∈ x0) (x3 x5 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x1) (proof)
Theorem explicit_Field_minus_clos^{explicit_Field_minus_clos} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x5 ∈ x0 (proof)
Theorem explicit_Field_minus_R^{explicit_Field_minus_R} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x5 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x1 (proof)
Theorem explicit_Field_minus_L^{explicit_Field_minus_L} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 (explicit_Field_minus x0 x1 x2 x3 x4 x5) x5 = x1 (proof)
Theorem explicit_Field_plus_cancelL^{explicit_Field_plus_cancelL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x6 = x3 x5 x7 ⟶ x6 = x7 (proof)
Theorem explicit_Field_plus_cancelR^{explicit_Field_plus_cancelR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x7 = x3 x6 x7 ⟶ x5 = x6 (proof)
Theorem explicit_Field_minus_invol^{explicit_Field_minus_invol} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x5 (proof)
Theorem explicit_Field_minus_one_In^{explicit_Field_minus_one_In} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x2 ∈ x0 (proof)
Theorem explicit_Field_zero_multR^{explicit_Field_zero_multR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5 x1 = x1 (proof)
Theorem explicit_Field_zero_multL^{explicit_Field_zero_multL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x1 x5 = x1 (proof)
Theorem explicit_Field_minus_mult^{explicit_Field_minus_mult} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x5 = x4 (explicit_Field_minus x0 x1 x2 x3 x4 x2) x5 (proof)
Theorem explicit_Field_minus_one_square^{explicit_Field_minus_one_square} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ x4 (explicit_Field_minus x0 x1 x2 x3 x4 x2) (explicit_Field_minus x0 x1 x2 x3 x4 x2) = x2 (proof)
Theorem explicit_Field_minus_square^{explicit_Field_minus_square} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 (explicit_Field_minus x0 x1 x2 x3 x4 x5) (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x4 x5 x5 (proof)
Definition iff^iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition explicit_OrderedField^{explicit_OrderedField} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . and (and (and (and (and (explicit_Field x0 x1 x2 x3 x4) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8)) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7))) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ or (x5 x6 x7) (x5 x7 x6))) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8))) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7))
Known and6I^and6I : ∀ x0 x1 x2 x3 x4 x5 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ and (and (and (and (and x0 x1) x2) x3) x4) x5
Theorem explicit_OrderedField_I^{explicit_OrderedField_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ or (x5 x6 x7) (x5 x7 x6)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7)) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 (proof)
Known and6E^and6E : ∀ x0 x1 x2 x3 x4 x5 : ο . and (and (and (and (and x0 x1) x2) x3) x4) x5 ⟶ ∀ x6 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6) ⟶ x6
Theorem explicit_OrderedField_E^{explicit_OrderedField_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x5 x7 x8 ⟶ x5 x8 x9 ⟶ x5 x7 x9) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ iff (and (x5 x7 x8) (x5 x8 x7)) (x7 = x8)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ or (x5 x7 x8) (x5 x8 x7)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x5 x7 x8 ⟶ x5 (x3 x7 x9) (x3 x8 x9)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x1 x7 ⟶ x5 x1 x8 ⟶ x5 x1 (x4 x7 x8)) ⟶ x6) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ x6 (proof)
Definition lt^lt := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 x7 . and (x5 x6 x7) (x6 = x7 ⟶ ∀ x8 : ο . x8)
Definition natOfOrderedField_p^{natOfOrderedField_p} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 . ∀ x7 : ι → ο . x7 x1 ⟶ (∀ x8 . x7 x8 ⟶ x7 (x3 x8 x2)) ⟶ x7 x6
Param Sep^Sep : ι → (ι → ο) → ι
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Theorem explicit_Nats_natOfOrderedField^{explicit_Nats_natOfOrderedField} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Nats (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) x1 (λ x6 . x3 x6 x2) (proof)
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Param ap^ap : ι → ι → ι
Definition explicit_Reals^{explicit_Reals} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . and (and (explicit_OrderedField x0 x1 x2 x3 x4 x5) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x7 (x4 x9 x6)) ⟶ x8) ⟶ x8)) (∀ x6 . x6 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x8 . x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x6 x8) (ap x7 x8)) (x5 (ap x6 x8) (ap x6 (x3 x8 x2)))) (x5 (ap x7 (x3 x8 x2)) (ap x7 x8))) ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ x0) (∀ x10 . x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x6 x10) x9) (x5 x9 (ap x7 x10))) ⟶ x8) ⟶ x8)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem explicit_Reals_I^{explicit_Reals_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x7 (x4 x9 x6)) ⟶ x8) ⟶ x8) ⟶ (∀ x6 . x6 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x8 . x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x6 x8) (ap x7 x8)) (x5 (ap x6 x8) (ap x6 (x3 x8 x2)))) (x5 (ap x7 (x3 x8 x2)) (ap x7 x8))) ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ x0) (∀ x10 . x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x6 x10) x9) (x5 x9 (ap x7 x10))) ⟶ x8) ⟶ x8) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5 (proof)
Known and3E^and3E : ∀ x0 x1 x2 : ο . and (and x0 x1) x2 ⟶ ∀ x3 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3) ⟶ x3
Theorem explicit_Reals_E^{explicit_Reals_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x7 ⟶ x5 x1 x8 ⟶ ∀ x9 : ο . (∀ x10 . and (x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x8 (x4 x10 x7)) ⟶ x9) ⟶ x9) ⟶ (∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x8 . x8 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x9 . x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x7 x9) (ap x8 x9)) (x5 (ap x7 x9) (ap x7 (x3 x9 x2)))) (x5 (ap x8 (x3 x9 x2)) (ap x8 x9))) ⟶ ∀ x9 : ο . (∀ x10 . and (x10 ∈ x0) (∀ x11 . x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x7 x11) x10) (x5 x10 (ap x8 x11))) ⟶ x9) ⟶ x9) ⟶ x6) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ x6 (proof)
Theorem explicit_Field_transfer^{explicit_Field_transfer} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 x7 . ∀ x8 x9 : ι → ι → ι . ∀ x10 : ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ bij x0 x5 x10 ⟶ x10 x1 = x6 ⟶ x10 x2 = x7 ⟶ (∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ x10 (x3 x11 x12) = x8 (x10 x11) (x10 x12)) ⟶ (∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x0 ⟶ x10 (x4 x11 x12) = x9 (x10 x11) (x10 x12)) ⟶ explicit_Field x5 x6 x7 x8 x9 (proof)
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem explicit_OrderedField_transfer^{explicit_OrderedField_transfer} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 x8 . ∀ x9 x10 : ι → ι → ι . ∀ x11 : ι → ι → ο . ∀ x12 : ι → ι . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ bij x0 x6 x12 ⟶ x12 x1 = x7 ⟶ x12 x2 = x8 ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ x12 (x3 x13 x14) = x9 (x12 x13) (x12 x14)) ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ x12 (x4 x13 x14) = x10 (x12 x13) (x12 x14)) ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ iff (x5 x13 x14) (x11 (x12 x13) (x12 x14))) ⟶ explicit_OrderedField x6 x7 x8 x9 x10 x11 (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known lam_Pi^lam_Pi : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3) ⟶ lam x0 x2 ∈ Pi x0 x1
Theorem explicit_Reals_transfer^{explicit_Reals_transfer} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 x8 . ∀ x9 x10 : ι → ι → ι . ∀ x11 : ι → ι → ο . ∀ x12 : ι → ι . explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ bij x0 x6 x12 ⟶ x12 x1 = x7 ⟶ x12 x2 = x8 ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ x12 (x3 x13 x14) = x9 (x12 x13) (x12 x14)) ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ x12 (x4 x13 x14) = x10 (x12 x13) (x12 x14)) ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ iff (x5 x13 x14) (x11 (x12 x13) (x12 x14))) ⟶ explicit_Reals x6 x7 x8 x9 x10 x11 (proof)
Definition explicit_Complex^{explicit_Complex} := λ x0 . λ x1 x2 : ι → ι . λ x3 x4 x5 . λ x6 x7 : ι → ι → ι . and (and (and (and (and (and (and (and (explicit_Field x0 x3 x4 x6 x7) (∀ x8 : ο . (∀ x9 : ι → ι → ο . explicit_Reals {x10 ∈ x0|x1 x10 = x10} x3 x4 x6 x7 x9 ⟶ x8) ⟶ x8)) (∀ x8 . x8 ∈ x0 ⟶ x2 x8 ∈ {x9 ∈ x0|x1 x9 = x9})) (x5 ∈ x0)) (∀ x8 . x8 ∈ x0 ⟶ x1 x8 ∈ x0)) (∀ x8 . x8 ∈ x0 ⟶ x2 x8 ∈ x0)) (∀ x8 . x8 ∈ x0 ⟶ x8 = x6 (x1 x8) (x7 x5 (x2 x8)))) (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x1 x8 = x1 x9 ⟶ x2 x8 = x2 x9 ⟶ x8 = x9)) (x6 (x7 x5 x5) x4 = x3)
Theorem explicit_Complex_I^{explicit_Complex_I} : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 x5 . ∀ x6 x7 : ι → ι → ι . explicit_Field x0 x3 x4 x6 x7 ⟶ (∀ x8 : ο . (∀ x9 : ι → ι → ο . explicit_Reals {x10 ∈ x0|x1 x10 = x10} x3 x4 x6 x7 x9 ⟶ x8) ⟶ x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x2 x8 ∈ {x9 ∈ x0|x1 x9 = x9}) ⟶ x5 ∈ x0 ⟶ (∀ x8 . x8 ∈ x0 ⟶ x1 x8 ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x2 x8 ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x8 = x6 (x1 x8) (x7 x5 (x2 x8))) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x1 x8 = x1 x9 ⟶ x2 x8 = x2 x9 ⟶ x8 = x9) ⟶ x6 (x7 x5 x5) x4 = x3 ⟶ explicit_Complex x0 x1 x2 x3 x4 x5 x6 x7 (proof)