Proofgold Address

Param SNo^SNo : ι → ο
Param SNoLe^SNoLe : ι → ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param SNoLt^SNoLt : ι → ι → ο
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known SNo_0^SNo_0 : SNo 0
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known neg_mul_SNo_Lt^{neg_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLt x0 0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x2 x1 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x0 x2)
Known nonneg_mul_SNo_Le^{nonneg_mul_SNo_Le} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x0 x2)
Theorem SNo_sqr_nonneg^{SNo_sqr_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 (mul_SNo x0 x0) (proof)
Known SNoLt_trichotomy_or_impred^{SNoLt_trichotomy_or_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (SNoLt x0 x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (SNoLt x1 x0 ⟶ x2) ⟶ x2
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known mul_SNo_neg_neg^{mul_SNo_neg_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt x1 0 ⟶ SNoLt 0 (mul_SNo x0 x1)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known mul_SNo_pos_pos^{mul_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (mul_SNo x0 x1)
Theorem SNo_zero_or_sqr_pos^{SNo_zero_or_sqr_pos} : ∀ x0 . SNo x0 ⟶ or (x0 = 0) (SNoLt 0 (mul_SNo x0 x0)) (proof)
Param CSNo^CSNo : ι → ο
Param CSNo_Re^CSNo_Re : ι → ι
Param SNo_pair^SNo_pair : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param CSNo_Im^CSNo_Im : ι → ι
Definition minus_CSNo^minus_CSNo := λ x0 . SNo_pair (minus_SNo (CSNo_Re x0)) (minus_SNo (CSNo_Im x0))
Known CSNo_Re2^CSNo_Re2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo_Re (SNo_pair x0 x1) = x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known CSNo_ReR^CSNo_ReR : ∀ x0 . CSNo x0 ⟶ SNo (CSNo_Re x0)
Known CSNo_ImR^CSNo_ImR : ∀ x0 . CSNo x0 ⟶ SNo (CSNo_Im x0)
Theorem 31582..^{minus_CSNo_CRe} : ∀ x0 . CSNo x0 ⟶ CSNo_Re (minus_CSNo x0) = minus_SNo (CSNo_Re x0) (proof)
Known CSNo_Im2^CSNo_Im2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo_Im (SNo_pair x0 x1) = x1
Theorem 4f721..^{minus_CSNo_CIm} : ∀ x0 . CSNo x0 ⟶ CSNo_Im (minus_CSNo x0) = minus_SNo (CSNo_Im x0) (proof)
Param add_SNo^add_SNo : ι → ι → ι
Definition add_CSNo^add_CSNo := λ x0 x1 . SNo_pair (add_SNo (CSNo_Re x0) (CSNo_Re x1)) (add_SNo (CSNo_Im x0) (CSNo_Im x1))
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Theorem 15fd0..^add_CSNo_CRe : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Re (add_CSNo x0 x1) = add_SNo (CSNo_Re x0) (CSNo_Re x1) (proof)
Theorem 2fac0..^add_CSNo_CIm : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Im (add_CSNo x0 x1) = add_SNo (CSNo_Im x0) (CSNo_Im x1) (proof)
Known CSNo_ReIm_split^{CSNo_ReIm_split} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Re x0 = CSNo_Re x1 ⟶ CSNo_Im x0 = CSNo_Im x1 ⟶ x0 = x1
Known CSNo_add_CSNo^{CSNo_add_CSNo} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (add_CSNo x0 x1)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Theorem 80a5b..^add_CSNo_com : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ add_CSNo x0 x1 = add_CSNo x1 x0 (proof)
Known f_eq_i^f_eq_i : ∀ x0 : ι → ι . ∀ x1 x2 . x1 = x2 ⟶ x0 x1 = x0 x2
Known add_SNo_assoc^{add_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo (add_SNo x0 x1) x2
Theorem b63e9..^{add_CSNo_assoc} : ∀ x0 x1 x2 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo x2 ⟶ add_CSNo x0 (add_CSNo x1 x2) = add_CSNo (add_CSNo x0 x1) x2 (proof)
Known add_SNo_com_4_inner_mid^{add_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (add_SNo x0 x2) (add_SNo x1 x3)
Theorem add_SNo_rotate_4_0312^{add_SNo_rotate_4_0312} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (add_SNo x0 x3) (add_SNo x1 x2) (proof)
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem c7e3d..^mul_CSNo_ReR : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ SNo (add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Re x1)) (minus_SNo (mul_SNo (CSNo_Im x0) (CSNo_Im x1)))) (proof)
Theorem 79ab2..^mul_CSNo_ImR : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ SNo (add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Im x1)) (mul_SNo (CSNo_Im x0) (CSNo_Re x1))) (proof)
Definition mul_CSNo^mul_CSNo := λ x0 x1 . SNo_pair (add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Re x1)) (minus_SNo (mul_SNo (CSNo_Im x0) (CSNo_Im x1)))) (add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Im x1)) (mul_SNo (CSNo_Im x0) (CSNo_Re x1)))
Theorem a8c42..^mul_CSNo_CRe : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Re (mul_CSNo x0 x1) = add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Re x1)) (minus_SNo (mul_SNo (CSNo_Im x0) (CSNo_Im x1))) (proof)
Theorem 2c425..^mul_CSNo_CIm : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Im (mul_CSNo x0 x1) = add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Im x1)) (mul_SNo (CSNo_Im x0) (CSNo_Re x1)) (proof)
Known CSNo_I^CSNo_I : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo (SNo_pair x0 x1)
Theorem d8721..^{CSNo_mul_CSNo} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (mul_CSNo x0 x1) (proof)
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Theorem 1e9ba..^mul_CSNo_com : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ mul_CSNo x0 x1 = mul_CSNo x1 x0 (proof)
Known mul_SNo_distrL^{mul_SNo_distrL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (add_SNo x1 x2) = add_SNo (mul_SNo x0 x1) (mul_SNo x0 x2)
Known mul_SNo_minus_distrR^{mul_SNo_minus_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
Known minus_add_SNo_distr^{minus_add_SNo_distr} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1)
Known SNo_mul_SNo_3^{SNo_mul_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (mul_SNo x0 (mul_SNo x1 x2))
Known mul_SNo_assoc^{mul_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo (mul_SNo x0 x1) x2
Known mul_SNo_minus_distrL^{mul_SNo_minus_distrL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) x1 = minus_SNo (mul_SNo x0 x1)
Known mul_SNo_distrR^{mul_SNo_distrR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (add_SNo x0 x1) x2 = add_SNo (mul_SNo x0 x2) (mul_SNo x1 x2)
Theorem 8912c..^{mul_CSNo_assoc} : ∀ x0 x1 x2 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo x2 ⟶ mul_CSNo x0 (mul_CSNo x1 x2) = mul_CSNo (mul_CSNo x0 x1) x2 (proof)
Known SNo_CSNo^SNo_CSNo : ∀ x0 . SNo x0 ⟶ CSNo x0
Theorem b5ed6..^CSNo_0 : CSNo 0 (proof)
Param ordsucc^ordsucc : ι → ι
Known SNo_1^SNo_1 : SNo 1
Theorem ca69e..^CSNo_1 : CSNo 1 (proof)
Known SNo_pair_0^SNo_pair_0 : ∀ x0 . SNo_pair x0 0 = x0
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Theorem 42258..^{mul_CSNo_oneL} : ∀ x0 . CSNo x0 ⟶ mul_CSNo 1 x0 = x0 (proof)
Theorem b904d..^{mul_CSNo_distrL} : ∀ x0 x1 x2 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo x2 ⟶ mul_CSNo x0 (add_CSNo x1 x2) = add_CSNo (mul_CSNo x0 x1) (mul_CSNo x0 x2) (proof)
Theorem b0c29.. : ∀ x0 x1 x2 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo x2 ⟶ mul_CSNo (add_CSNo x0 x1) x2 = add_CSNo (mul_CSNo x0 x2) (mul_CSNo x1 x2) (proof)
Known SNo_Re^SNo_Re : ∀ x0 . SNo x0 ⟶ CSNo_Re x0 = x0
Known SNo_Im^SNo_Im : ∀ x0 . SNo x0 ⟶ CSNo_Im x0 = 0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Theorem 15de6..^{mul_SNo_mul_CSNo} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_CSNo x0 x1 (proof)
Definition Complex_i^Complex_i := SNo_pair 0 1
Known SNo_Complex_i^{CSNo_Complex_i} : CSNo Complex_i
Known CSNo_minus_CSNo^{CSNo_minus_CSNo} : ∀ x0 . CSNo x0 ⟶ CSNo (minus_CSNo x0)
Known Re_i^Re_i : CSNo_Re Complex_i = 0
Known Im_i^Im_i : CSNo_Im Complex_i = 1
Known Re_1^Re_1 : CSNo_Re 1 = 1
Known Im_1^Im_1 : CSNo_Im 1 = 0
Theorem b65ee..^{Complex_i_sqr} : mul_CSNo Complex_i Complex_i = minus_CSNo 1 (proof)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known mul_SNo_com_3_0_1^{mul_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo x1 (mul_SNo x0 x2)
Known add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Theorem f9fad..^{CSNo_relative_recip} : ∀ x0 . CSNo x0 ⟶ ∀ x1 . SNo x1 ⟶ mul_SNo (add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Re x0)) (mul_SNo (CSNo_Im x0) (CSNo_Im x0))) x1 = 1 ⟶ mul_CSNo x0 (add_CSNo (mul_CSNo x1 (CSNo_Re x0)) (minus_CSNo (mul_CSNo Complex_i (mul_CSNo x1 (CSNo_Im x0))))) = 1 (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param real^real : ι
Param ap^ap : ι → ι → ι
Definition complex^complex := {SNo_pair (ap x0 0) (ap x0 1)|x0 ∈ setprod real real}
Param If_i^If_i : ο → ι → ι → ι
Known tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known tuple_2_setprod^{tuple_2_setprod} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ setprod x0 x1
Theorem complex_I^complex_I : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ SNo_pair x0 x1 ∈ complex (proof)
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Known ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Theorem complex_E^complex_E : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ real ⟶ ∀ x3 . x3 ∈ real ⟶ x0 = SNo_pair x2 x3 ⟶ x1) ⟶ x1 (proof)
Known real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Theorem complex_CSNo^complex_CSNo : ∀ x0 . x0 ∈ complex ⟶ CSNo x0 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known real_0^real_0 : 0 ∈ real
Theorem real_complex^real_complex : real ⊆ complex (proof)
Theorem complex_0^complex_0 : 0 ∈ complex (proof)
Known real_1^real_1 : 1 ∈ real
Theorem complex_1^complex_1 : 1 ∈ complex (proof)
Theorem complex_i^complex_i : Complex_i ∈ complex (proof)
Theorem complex_Re_eq^{complex_Re_eq} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ CSNo_Re (SNo_pair x0 x1) = x0 (proof)
Theorem complex_Im_eq^{complex_Im_eq} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ CSNo_Im (SNo_pair x0 x1) = x1 (proof)
Known CSNo_ReIm^CSNo_ReIm : ∀ x0 . CSNo x0 ⟶ x0 = SNo_pair (CSNo_Re x0) (CSNo_Im x0)
Theorem a6f8f.. : ∀ x0 . x0 ∈ complex ⟶ x0 = SNo_pair (CSNo_Re x0) (CSNo_Im x0) (proof)
Theorem complex_Re_real^{complex_Re_real} : ∀ x0 . x0 ∈ complex ⟶ CSNo_Re x0 ∈ real (proof)
Theorem complex_Im_real^{complex_Im_real} : ∀ x0 . x0 ∈ complex ⟶ CSNo_Im x0 ∈ real (proof)
Theorem complex_ReIm_split^{complex_ReIm_split} : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ CSNo_Re x0 = CSNo_Re x1 ⟶ CSNo_Im x0 = CSNo_Im x1 ⟶ x0 = x1 (proof)
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Theorem complex_minus_CSNo^{complex_minus_CSNo} : ∀ x0 . x0 ∈ complex ⟶ minus_CSNo x0 ∈ complex (proof)
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Theorem complex_add_CSNo^{complex_add_CSNo} : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ add_CSNo x0 x1 ∈ complex (proof)
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Theorem complex_mul_CSNo^{complex_mul_CSNo} : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ mul_CSNo x0 x1 ∈ complex (proof)
Theorem real_Re_eq^real_Re_eq : ∀ x0 . x0 ∈ real ⟶ CSNo_Re x0 = x0 (proof)
Theorem real_Im_eq^real_Im_eq : ∀ x0 . x0 ∈ real ⟶ CSNo_Im x0 = 0 (proof)
Theorem mul_i_real_eq^{mul_i_real_eq} : ∀ x0 . x0 ∈ real ⟶ mul_CSNo Complex_i x0 = SNo_pair 0 x0 (proof)
Theorem real_Re_i_eq^real_Re_i_eq : ∀ x0 . x0 ∈ real ⟶ CSNo_Re (mul_CSNo Complex_i x0) = 0 (proof)
Theorem real_Im_i_eq^real_Im_i_eq : ∀ x0 . x0 ∈ real ⟶ CSNo_Im (mul_CSNo Complex_i x0) = x0 (proof)
Theorem complex_eta^complex_eta : ∀ x0 . x0 ∈ complex ⟶ x0 = add_CSNo (CSNo_Re x0) (mul_CSNo Complex_i (CSNo_Im x0)) (proof)
Theorem dfa13.. : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_CSNo x0 (mul_CSNo Complex_i x1) ∈ complex (proof)
Theorem 894e6.. : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ real ⟶ ∀ x3 . x3 ∈ real ⟶ x0 = add_CSNo x2 (mul_CSNo Complex_i x3) ⟶ x1) ⟶ x1 (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known nonzero_real_recip_ex^{nonzero_real_recip_ex} : ∀ x0 . x0 ∈ real ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ real) (mul_SNo x0 x2 = 1) ⟶ x1) ⟶ x1
Known Re_0^Re_0 : CSNo_Re 0 = 0
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known add_SNo_Le2^add_SNo_Le2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x0 x2)
Known Im_0^Im_0 : CSNo_Im 0 = 0
Known add_SNo_Le1^add_SNo_Le1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x1)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem nonzero_complex_recip_ex^{nonzero_complex_recip_ex} : ∀ x0 . x0 ∈ complex ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ complex) (mul_CSNo x0 x2 = 1) ⟶ x1) ⟶ x1 (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
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 complex_real_set_eq^{complex_real_set_eq} : real = {x1 ∈ complex|CSNo_Re x1 = x1} (proof)
Param explicit_Field^{explicit_Field} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Known 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
Known add_CSNo_0L^add_CSNo_0L : ∀ x0 . CSNo x0 ⟶ add_CSNo 0 x0 = x0
Known add_CSNo_minus_CSNo_rinv^{add_CSNo_minus_CSNo_rinv} : ∀ x0 . CSNo x0 ⟶ add_CSNo x0 (minus_CSNo x0) = 0
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Theorem 881af.. : explicit_Field complex 0 1 add_CSNo mul_CSNo (proof)
Param explicit_Complex^{explicit_Complex} : ι → (ι → ι) → (ι → ι) → ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Known 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
Param bij^bij : ι → ι → (ι → ι) → ο
Param iff^iff : ο → ο → ο
Known 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
Known explicit_Reals_real : explicit_Reals real 0 1 add_SNo mul_SNo SNoLe
Known bij_id^bij_id : ∀ x0 . bij x0 x0 (λ x1 . x1)
Known add_SNo_add_CSNo^{add_SNo_add_CSNo} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_CSNo x0 x1
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Known add_CSNo_minus_CSNo_linv^{add_CSNo_minus_CSNo_linv} : ∀ x0 . CSNo x0 ⟶ add_CSNo (minus_CSNo x0) x0 = 0
Theorem 73d02.. : explicit_Complex complex CSNo_Re CSNo_Im 0 1 Complex_i add_CSNo mul_CSNo (proof)