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Proofgold Proof
pf
Claim L0:
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Claim L1:
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Claim L2:
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Claim L3:
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Apply OSNo_proj0proj1_split with
mul_OSNo
Quaternion_j
Octonion_i5
,
minus_OSNo
Octonion_i3
leaving 4 subgoals.
The subproof is completed by applying L3.
Apply OSNo_minus_OSNo with
Octonion_i3
.
The subproof is completed by applying L2.
Apply minus_OSNo_proj0 with
Octonion_i3
,
λ x0 x1 .
OSNo_proj0
(
mul_OSNo
Quaternion_j
Octonion_i5
)
=
x1
leaving 2 subgoals.
The subproof is completed by applying L2.
Apply OSNo_p0_i3 with
λ x0 x1 .
OSNo_proj0
(
mul_OSNo
Quaternion_j
Octonion_i5
)
=
minus_HSNo
x1
.
Apply mul_OSNo_proj0 with
Quaternion_j
,
Octonion_i5
,
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_i5 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
OSNo_proj0
Quaternion_j
)
x1
)
(
minus_HSNo
(
mul_HSNo
(
conj_HSNo
(
OSNo_proj1
Octonion_i5
)
)
(
OSNo_proj1
Quaternion_j
)
)
)
=
minus_HSNo
0
.
Apply OSNo_p1_i5 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
(
minus_HSNo
Quaternion_k
)
)
(
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
(
minus_HSNo
Quaternion_k
)
)
x1
)
)
=
minus_HSNo
0
.
Apply mul_HSNo_0R with
Quaternion_j
,
λ x0 x1 .
add_HSNo
x1
(
minus_HSNo
(
mul_HSNo
(
conj_HSNo
(
minus_HSNo
Quaternion_k
)
)
0
)
)
=
minus_HSNo
0
leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_j.
Apply mul_HSNo_0R with
conj_HSNo
(
minus_HSNo
Quaternion_k
)
,
λ x0 x1 .
add_HSNo
0
(
minus_HSNo
x1
)
=
minus_HSNo
0
leaving 2 subgoals.
Apply HSNo_conj_HSNo with
minus_HSNo
Quaternion_k
.
Apply HSNo_minus_HSNo with
Quaternion_k
.
The subproof is completed by applying HSNo_Quaternion_k.
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_i3
,
λ x0 x1 .
OSNo_proj1
(
mul_OSNo
Quaternion_j
Octonion_i5
)
=
x1
leaving 2 subgoals.
The subproof is completed by applying L2.
Apply OSNo_p1_i3 with
λ x0 x1 .
OSNo_proj1
(
mul_OSNo
Quaternion_j
Octonion_i5
)
=
minus_HSNo
x1
.
Apply mul_OSNo_proj1 with
Quaternion_j
,
Octonion_i5
,
λ x0 x1 .
x1
=
minus_HSNo
(
minus_HSNo
Complex_i
)
leaving 3 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying L0.
Apply OSNo_p0_i5 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
OSNo_proj1
Octonion_i5
)
(
OSNo_proj0
Quaternion_j
)
)
(
mul_HSNo
(
OSNo_proj1
Quaternion_j
)
(
conj_HSNo
x1
)
)
=
minus_HSNo
(
minus_HSNo
Complex_i
)
.
Apply OSNo_p1_i5 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
Complex_i
)
.
Apply OSNo_p0_j with
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
x1
)
(
mul_HSNo
(
OSNo_proj1
Quaternion_j
)
(
conj_HSNo
0
)
)
=
minus_HSNo
(
minus_HSNo
Complex_i
)
.
Apply OSNo_p1_j with
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
Quaternion_j
)
(
mul_HSNo
x1
(
conj_HSNo
0
)
)
=
minus_HSNo
(
minus_HSNo
Complex_i
)
.
Apply conj_HSNo_id_SNo with
0
,
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
Quaternion_j
)
(
mul_HSNo
0
x1
)
=
minus_HSNo
(
minus_HSNo
Complex_i
)
leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply mul_HSNo_0L with
0
,
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
Quaternion_j
)
x1
=
minus_HSNo
(
minus_HSNo
Complex_i
)
leaving 2 subgoals.
The subproof is completed by applying HSNo_0.
Apply minus_mul_HSNo_distrL with
Quaternion_k
,
Quaternion_j
,
λ x0 x1 .
add_HSNo
x1
0
=
minus_HSNo
(
minus_HSNo
Complex_i
)
leaving 3 subgoals.
The subproof is completed by applying HSNo_Quaternion_k.
The subproof is completed by applying HSNo_Quaternion_j.
Apply Quaternion_k_j with
λ x0 x1 .
add_HSNo
(
minus_HSNo
x1
)
0
=
minus_HSNo
(
minus_HSNo
Complex_i
)
.
Apply minus_HSNo_invol with
Complex_i
,
λ x0 x1 .
add_HSNo
x1
0
=
x1
leaving 2 subgoals.
The subproof is completed by applying HSNo_Complex_i.
Apply add_HSNo_0R with
Complex_i
.
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