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Proofgold Proof

pf
Claim L0: OSNo Octonion_i0
The subproof is completed by applying OSNo_Octonion_i0.
Claim L1: OSNo Quaternion_j
The subproof is completed by applying OSNo_Quaternion_j.
Claim L2: OSNo Octonion_i6
The subproof is completed by applying OSNo_Octonion_i6.
Claim L3: OSNo (mul_OSNo Quaternion_j Octonion_i0)
Apply OSNo_mul_OSNo with Quaternion_j, Octonion_i0 leaving 2 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying L0.
Apply OSNo_proj0proj1_split with mul_OSNo Quaternion_j Octonion_i0, minus_OSNo Octonion_i6 leaving 4 subgoals.
The subproof is completed by applying L3.
Apply OSNo_minus_OSNo with Octonion_i6.
The subproof is completed by applying L2.
Apply minus_OSNo_proj0 with Octonion_i6, λ x0 x1 . OSNo_proj0 (mul_OSNo Quaternion_j Octonion_i0) = x1 leaving 2 subgoals.
The subproof is completed by applying L2.
Apply OSNo_p0_i6 with λ x0 x1 . OSNo_proj0 (mul_OSNo Quaternion_j Octonion_i0) = minus_HSNo x1.
Apply mul_OSNo_proj0 with Quaternion_j, Octonion_i0, λ x0 x1 . x1 = minus_HSNo 0 leaving 3 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying L0.
Apply OSNo_p0_i0 with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj0 Quaternion_j) x1) (minus_HSNo (mul_HSNo (conj_HSNo (OSNo_proj1 Octonion_i0)) (OSNo_proj1 Quaternion_j))) = minus_HSNo 0.
Apply OSNo_p1_i0 with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj0 Quaternion_j) 0) (minus_HSNo (mul_HSNo (conj_HSNo x1) (OSNo_proj1 Quaternion_j))) = minus_HSNo 0.
Apply OSNo_p0_j with λ x0 x1 . add_HSNo (mul_HSNo x1 0) (minus_HSNo (mul_HSNo (conj_HSNo 1) (OSNo_proj1 Quaternion_j))) = minus_HSNo 0.
Apply OSNo_p1_j with λ x0 x1 . add_HSNo (mul_HSNo Quaternion_j 0) (minus_HSNo (mul_HSNo (conj_HSNo 1) x1)) = minus_HSNo 0.
Apply mul_HSNo_0R with Quaternion_j, λ x0 x1 . add_HSNo x1 (minus_HSNo (mul_HSNo (conj_HSNo 1) 0)) = minus_HSNo 0 leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
Apply mul_HSNo_0R with conj_HSNo 1, λ x0 x1 . add_HSNo 0 (minus_HSNo x1) = minus_HSNo 0 leaving 2 subgoals.
Apply HSNo_conj_HSNo with 1.
The subproof is completed by applying HSNo_1.
Apply minus_HSNo_0 with λ x0 x1 . add_HSNo 0 x1 = x1.
Apply add_HSNo_0L with 0.
The subproof is completed by applying HSNo_0.
Apply minus_OSNo_proj1 with Octonion_i6, λ x0 x1 . OSNo_proj1 (mul_OSNo Quaternion_j Octonion_i0) = x1 leaving 2 subgoals.
The subproof is completed by applying L2.
Apply OSNo_p1_i6 with λ x0 x1 . OSNo_proj1 (mul_OSNo Quaternion_j Octonion_i0) = minus_HSNo x1.
Apply mul_OSNo_proj1 with Quaternion_j, Octonion_i0, λ x0 x1 . x1 = minus_HSNo (minus_HSNo Quaternion_j) leaving 3 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying L0.
Apply OSNo_p0_i0 with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj1 Octonion_i0) (OSNo_proj0 Quaternion_j)) (mul_HSNo (OSNo_proj1 Quaternion_j) (conj_HSNo x1)) = minus_HSNo (minus_HSNo Quaternion_j).
Apply OSNo_p1_i0 with λ x0 x1 . add_HSNo (mul_HSNo x1 (OSNo_proj0 Quaternion_j)) (mul_HSNo (OSNo_proj1 Quaternion_j) (conj_HSNo 0)) = minus_HSNo (minus_HSNo Quaternion_j).
Apply OSNo_p0_j with λ x0 x1 . add_HSNo (mul_HSNo 1 x1) (mul_HSNo (OSNo_proj1 Quaternion_j) (conj_HSNo 0)) = minus_HSNo (minus_HSNo Quaternion_j).
Apply OSNo_p1_j with λ x0 x1 . add_HSNo (mul_HSNo 1 Quaternion_j) (mul_HSNo x1 (conj_HSNo 0)) = minus_HSNo (minus_HSNo Quaternion_j).
Apply conj_HSNo_id_SNo with 0, λ x0 x1 . add_HSNo (mul_HSNo 1 Quaternion_j) (mul_HSNo 0 x1) = minus_HSNo (minus_HSNo Quaternion_j) leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply mul_HSNo_0L with 0, λ x0 x1 . add_HSNo (mul_HSNo 1 Quaternion_j) x1 = minus_HSNo (minus_HSNo Quaternion_j) leaving 2 subgoals.
The subproof is completed by applying HSNo_0.
Apply minus_HSNo_invol with Quaternion_j, λ x0 x1 . add_HSNo (mul_HSNo 1 Quaternion_j) 0 = x1 leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
Apply mul_HSNo_1L with Quaternion_j, λ x0 x1 . add_HSNo x1 0 = Quaternion_j leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
Apply add_HSNo_0R with Quaternion_j.
The subproof is completed by applying HSNo_Quaternion_j.
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