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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
Octonion_i0
Octonion_i5
,
minus_OSNo
Quaternion_k
leaving 4 subgoals.
The subproof is completed by applying L3.
Apply OSNo_minus_OSNo with
Quaternion_k
.
The subproof is completed by applying L2.
Apply minus_OSNo_proj0 with
Quaternion_k
,
λ x0 x1 .
OSNo_proj0
(
mul_OSNo
Octonion_i0
Octonion_i5
)
=
x1
leaving 2 subgoals.
The subproof is completed by applying L2.
Apply OSNo_p0_k with
λ x0 x1 .
OSNo_proj0
(
mul_OSNo
Octonion_i0
Octonion_i5
)
=
minus_HSNo
x1
.
Apply mul_OSNo_proj0 with
Octonion_i0
,
Octonion_i5
,
λ x0 x1 .
x1
=
minus_HSNo
Quaternion_k
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
Octonion_i0
)
x1
)
(
minus_HSNo
(
mul_HSNo
(
conj_HSNo
(
OSNo_proj1
Octonion_i5
)
)
(
OSNo_proj1
Octonion_i0
)
)
)
=
minus_HSNo
Quaternion_k
.
Apply OSNo_p1_i5 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
OSNo_proj0
Octonion_i0
)
0
)
(
minus_HSNo
(
mul_HSNo
(
conj_HSNo
x1
)
(
OSNo_proj1
Octonion_i0
)
)
)
=
minus_HSNo
Quaternion_k
.
Apply OSNo_p0_i0 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
x1
0
)
(
minus_HSNo
(
mul_HSNo
(
conj_HSNo
(
minus_HSNo
Quaternion_k
)
)
(
OSNo_proj1
Octonion_i0
)
)
)
=
minus_HSNo
Quaternion_k
.
Apply OSNo_p1_i0 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
0
0
)
(
minus_HSNo
(
mul_HSNo
(
conj_HSNo
(
minus_HSNo
Quaternion_k
)
)
x1
)
)
=
minus_HSNo
Quaternion_k
.
Apply mul_HSNo_0L with
0
,
λ x0 x1 .
add_HSNo
x1
(
minus_HSNo
(
mul_HSNo
(
conj_HSNo
(
minus_HSNo
Quaternion_k
)
)
1
)
)
=
minus_HSNo
Quaternion_k
leaving 2 subgoals.
The subproof is completed by applying HSNo_0.
Apply conj_minus_HSNo with
Quaternion_k
,
λ x0 x1 .
add_HSNo
0
(
minus_HSNo
(
mul_HSNo
x1
1
)
)
=
minus_HSNo
Quaternion_k
leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_k.
Apply conj_HSNo_k with
λ x0 x1 .
add_HSNo
0
(
minus_HSNo
(
mul_HSNo
(
minus_HSNo
x1
)
1
)
)
=
minus_HSNo
Quaternion_k
.
Apply minus_HSNo_invol with
Quaternion_k
,
λ x0 x1 .
add_HSNo
0
(
minus_HSNo
(
mul_HSNo
x1
1
)
)
=
minus_HSNo
Quaternion_k
leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_k.
Apply mul_HSNo_1R with
Quaternion_k
,
λ x0 x1 .
add_HSNo
0
(
minus_HSNo
x1
)
=
minus_HSNo
Quaternion_k
leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_k.
Apply add_HSNo_0L with
minus_HSNo
Quaternion_k
.
Apply HSNo_minus_HSNo with
Quaternion_k
.
The subproof is completed by applying HSNo_Quaternion_k.
Apply minus_OSNo_proj1 with
Quaternion_k
,
λ x0 x1 .
OSNo_proj1
(
mul_OSNo
Octonion_i0
Octonion_i5
)
=
x1
leaving 2 subgoals.
The subproof is completed by applying L2.
Apply OSNo_p1_k with
λ x0 x1 .
OSNo_proj1
(
mul_OSNo
Octonion_i0
Octonion_i5
)
=
minus_HSNo
x1
.
Apply mul_OSNo_proj1 with
Octonion_i0
,
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_proj1
Octonion_i5
)
(
OSNo_proj0
Octonion_i0
)
)
(
mul_HSNo
(
OSNo_proj1
Octonion_i0
)
(
conj_HSNo
x1
)
)
=
minus_HSNo
0
.
Apply OSNo_p1_i5 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
x1
(
OSNo_proj0
Octonion_i0
)
)
(
mul_HSNo
(
OSNo_proj1
Octonion_i0
)
(
conj_HSNo
0
)
)
=
minus_HSNo
0
.
Apply OSNo_p0_i0 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
x1
)
(
mul_HSNo
(
OSNo_proj1
Octonion_i0
)
(
conj_HSNo
0
)
)
=
minus_HSNo
0
.
Apply OSNo_p1_i0 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
0
)
(
mul_HSNo
x1
(
conj_HSNo
0
)
)
=
minus_HSNo
0
.
Apply conj_HSNo_id_SNo with
0
,
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
0
)
(
mul_HSNo
1
x1
)
=
minus_HSNo
0
leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply minus_HSNo_0 with
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
0
)
(
mul_HSNo
1
0
)
=
x1
.
Apply mul_HSNo_0R with
1
,
λ x0 x1 .
add_HSNo
(
mul_HSNo
(
minus_HSNo
Quaternion_k
)
0
)
x1
=
0
leaving 2 subgoals.
The subproof is completed by applying HSNo_1.
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