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Proofgold Proof

pf
Claim L0: OSNo Quaternion_j
The subproof is completed by applying OSNo_Quaternion_j.
Claim L1: OSNo Quaternion_k
The subproof is completed by applying OSNo_Quaternion_k.
Claim L2: OSNo Complex_i
The subproof is completed by applying OSNo_Complex_i.
Claim L3: OSNo (mul_OSNo Quaternion_j Quaternion_k)
Apply OSNo_mul_OSNo with Quaternion_j, Quaternion_k leaving 2 subgoals.
The subproof is completed by applying L0.
The subproof is completed by applying L1.
Apply OSNo_proj0proj1_split with mul_OSNo Quaternion_j Quaternion_k, Complex_i leaving 4 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L2.
Apply OSNo_p0_i with λ x0 x1 . OSNo_proj0 (mul_OSNo Quaternion_j Quaternion_k) = x1.
Apply mul_OSNo_proj0 with Quaternion_j, Quaternion_k, λ x0 x1 . x1 = Complex_i leaving 3 subgoals.
The subproof is completed by applying L0.
The subproof is completed by applying L1.
Apply OSNo_p0_j with λ x0 x1 . add_HSNo (mul_HSNo x1 (OSNo_proj0 Quaternion_k)) (minus_HSNo (mul_HSNo (conj_HSNo (OSNo_proj1 Quaternion_k)) (OSNo_proj1 Quaternion_j))) = Complex_i.
Apply OSNo_p1_j with λ x0 x1 . add_HSNo (mul_HSNo Quaternion_j (OSNo_proj0 Quaternion_k)) (minus_HSNo (mul_HSNo (conj_HSNo (OSNo_proj1 Quaternion_k)) x1)) = Complex_i.
Apply OSNo_p0_k with λ x0 x1 . add_HSNo (mul_HSNo Quaternion_j x1) (minus_HSNo (mul_HSNo (conj_HSNo (OSNo_proj1 Quaternion_k)) 0)) = Complex_i.
Apply OSNo_p1_k with λ x0 x1 . add_HSNo (mul_HSNo Quaternion_j Quaternion_k) (minus_HSNo (mul_HSNo (conj_HSNo x1) 0)) = Complex_i.
Apply conj_HSNo_id_SNo with 0, λ x0 x1 . add_HSNo (mul_HSNo Quaternion_j Quaternion_k) (minus_HSNo (mul_HSNo x1 0)) = Complex_i leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply mul_HSNo_0R with 0, λ x0 x1 . add_HSNo (mul_HSNo Quaternion_j Quaternion_k) (minus_HSNo x1) = Complex_i leaving 2 subgoals.
The subproof is completed by applying HSNo_0.
Apply minus_HSNo_0 with λ x0 x1 . add_HSNo (mul_HSNo Quaternion_j Quaternion_k) x1 = Complex_i.
Apply add_HSNo_0R with mul_HSNo Quaternion_j Quaternion_k, λ x0 x1 . x1 = Complex_i leaving 2 subgoals.
Apply HSNo_mul_HSNo with Quaternion_j, Quaternion_k leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
The subproof is completed by applying HSNo_Quaternion_k.
The subproof is completed by applying Quaternion_j_k.
Apply OSNo_p1_i with λ x0 x1 . OSNo_proj1 (mul_OSNo Quaternion_j Quaternion_k) = x1.
Apply mul_OSNo_proj1 with Quaternion_j, Quaternion_k, λ x0 x1 . x1 = 0 leaving 3 subgoals.
The subproof is completed by applying L0.
The subproof is completed by applying L1.
Apply OSNo_p0_j with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj1 Quaternion_k) x1) (mul_HSNo (OSNo_proj1 Quaternion_j) (conj_HSNo (OSNo_proj0 Quaternion_k))) = 0.
Apply OSNo_p1_j with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj1 Quaternion_k) Quaternion_j) (mul_HSNo x1 (conj_HSNo (OSNo_proj0 Quaternion_k))) = 0.
Apply OSNo_p0_k with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj1 Quaternion_k) Quaternion_j) (mul_HSNo 0 (conj_HSNo x1)) = 0.
Apply OSNo_p1_k with λ x0 x1 . add_HSNo (mul_HSNo x1 Quaternion_j) (mul_HSNo 0 (conj_HSNo Quaternion_k)) = 0.
Apply mul_HSNo_0L with conj_HSNo Quaternion_k, λ x0 x1 . add_HSNo (mul_HSNo 0 Quaternion_j) x1 = 0 leaving 2 subgoals.
Apply HSNo_conj_HSNo with Quaternion_k.
The subproof is completed by applying HSNo_Quaternion_k.
Apply mul_HSNo_0L with Quaternion_j, λ x0 x1 . add_HSNo x1 0 = 0 leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
Apply add_HSNo_0L with 0.
The subproof is completed by applying HSNo_0.
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