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Proofgold Proof
pf
Claim L0:
HSNo
(
mul_HSNo
Quaternion_j
Complex_i
)
Apply HSNo_mul_HSNo with
Quaternion_j
,
Complex_i
leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
The subproof is completed by applying HSNo_Complex_i.
Apply HSNo_proj0proj1_split with
mul_HSNo
Quaternion_j
Complex_i
,
minus_HSNo
Quaternion_k
leaving 4 subgoals.
The subproof is completed by applying L0.
Apply HSNo_minus_HSNo with
Quaternion_k
.
The subproof is completed by applying HSNo_Quaternion_k.
Apply minus_HSNo_proj0 with
Quaternion_k
,
λ x0 x1 .
HSNo_proj0
(
mul_HSNo
Quaternion_j
Complex_i
)
=
x1
leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_k.
Apply HSNo_p0_k with
λ x0 x1 .
HSNo_proj0
(
mul_HSNo
Quaternion_j
Complex_i
)
=
minus_CSNo
x1
.
Apply mul_HSNo_proj0 with
Quaternion_j
,
Complex_i
,
λ x0 x1 .
x1
=
minus_CSNo
0
leaving 3 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
The subproof is completed by applying HSNo_Complex_i.
Apply HSNo_p0_j with
λ x0 x1 .
add_CSNo
(
mul_CSNo
x1
(
HSNo_proj0
Complex_i
)
)
(
minus_CSNo
(
mul_CSNo
(
conj_CSNo
(
HSNo_proj1
Complex_i
)
)
(
HSNo_proj1
Quaternion_j
)
)
)
=
minus_CSNo
0
.
Apply HSNo_p1_j with
λ x0 x1 .
add_CSNo
(
mul_CSNo
0
(
HSNo_proj0
Complex_i
)
)
(
minus_CSNo
(
mul_CSNo
(
conj_CSNo
(
HSNo_proj1
Complex_i
)
)
x1
)
)
=
minus_CSNo
0
.
Apply HSNo_p0_i with
λ x0 x1 .
add_CSNo
(
mul_CSNo
0
x1
)
(
minus_CSNo
(
mul_CSNo
(
conj_CSNo
(
HSNo_proj1
Complex_i
)
)
1
)
)
=
minus_CSNo
0
.
Apply HSNo_p1_i with
λ x0 x1 .
add_CSNo
(
mul_CSNo
0
Complex_i
)
(
minus_CSNo
(
mul_CSNo
(
conj_CSNo
x1
)
1
)
)
=
minus_CSNo
0
.
Apply conj_CSNo_id_SNo with
0
,
λ x0 x1 .
add_CSNo
(
mul_CSNo
0
Complex_i
)
(
minus_CSNo
(
mul_CSNo
x1
1
)
)
=
minus_CSNo
0
leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply mul_CSNo_0L with
Complex_i
,
λ x0 x1 .
add_CSNo
x1
(
minus_CSNo
(
mul_CSNo
0
1
)
)
=
minus_CSNo
0
leaving 2 subgoals.
The subproof is completed by applying CSNo_Complex_i.
Apply mul_CSNo_0L with
1
,
λ x0 x1 .
add_CSNo
0
(
minus_CSNo
x1
)
=
minus_CSNo
0
leaving 2 subgoals.
The subproof is completed by applying CSNo_1.
Apply add_CSNo_0L with
minus_CSNo
0
.
Apply CSNo_minus_CSNo with
0
.
The subproof is completed by applying CSNo_0.
Apply minus_HSNo_proj1 with
Quaternion_k
,
λ x0 x1 .
HSNo_proj1
(
mul_HSNo
Quaternion_j
Complex_i
)
=
x1
leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_k.
Apply HSNo_p1_k with
λ x0 x1 .
HSNo_proj1
(
mul_HSNo
Quaternion_j
Complex_i
)
=
minus_CSNo
x1
.
Apply mul_HSNo_proj1 with
Quaternion_j
,
Complex_i
,
λ x0 x1 .
x1
=
minus_CSNo
Complex_i
leaving 3 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
The subproof is completed by applying HSNo_Complex_i.
Apply HSNo_p0_j with
λ x0 x1 .
add_CSNo
(
mul_CSNo
(
HSNo_proj1
Complex_i
)
x1
)
(
mul_CSNo
(
HSNo_proj1
Quaternion_j
)
(
conj_CSNo
(
HSNo_proj0
Complex_i
)
)
)
=
minus_CSNo
Complex_i
.
Apply HSNo_p1_j with
λ x0 x1 .
add_CSNo
(
mul_CSNo
(
HSNo_proj1
Complex_i
)
0
)
(
mul_CSNo
x1
(
conj_CSNo
(
HSNo_proj0
Complex_i
)
)
)
=
minus_CSNo
Complex_i
.
Apply HSNo_p0_i with
λ x0 x1 .
add_CSNo
(
mul_CSNo
(
HSNo_proj1
Complex_i
)
0
)
(
mul_CSNo
1
(
conj_CSNo
x1
)
)
=
minus_CSNo
Complex_i
.
Apply HSNo_p1_i with
λ x0 x1 .
add_CSNo
(
mul_CSNo
x1
0
)
(
mul_CSNo
1
(
conj_CSNo
Complex_i
)
)
=
minus_CSNo
Complex_i
.
Apply mul_CSNo_0R with
0
,
λ x0 x1 .
add_CSNo
x1
(
mul_CSNo
1
(
conj_CSNo
Complex_i
)
)
=
minus_CSNo
Complex_i
leaving 2 subgoals.
The subproof is completed by applying CSNo_0.
Apply mul_CSNo_1L with
conj_CSNo
Complex_i
,
λ x0 x1 .
add_CSNo
0
x1
=
minus_CSNo
Complex_i
leaving 2 subgoals.
Apply CSNo_conj_CSNo with
Complex_i
.
The subproof is completed by applying CSNo_Complex_i.
Apply conj_CSNo_i with
λ x0 x1 .
add_CSNo
0
x1
=
minus_CSNo
Complex_i
.
Apply add_CSNo_0L with
minus_CSNo
Complex_i
.
Apply CSNo_minus_CSNo with
Complex_i
.
The subproof is completed by applying CSNo_Complex_i.
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