Step |
Hyp |
Ref |
Expression |
1 |
|
ply1scl.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
ply1scl.a |
⊢ 𝐴 = ( algSc ‘ 𝑃 ) |
3 |
|
coe1scl.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
coe1scl.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( var1 ‘ 𝑅 ) = ( var1 ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ 𝑃 ) |
7 |
|
eqid |
⊢ ( mulGrp ‘ 𝑃 ) = ( mulGrp ‘ 𝑃 ) |
8 |
|
eqid |
⊢ ( .g ‘ ( mulGrp ‘ 𝑃 ) ) = ( .g ‘ ( mulGrp ‘ 𝑃 ) ) |
9 |
3 1 5 6 7 8 2
|
ply1scltm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( 𝐴 ‘ 𝑋 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) |
10 |
9
|
fveq2d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( coe1 ‘ ( 𝐴 ‘ 𝑋 ) ) = ( coe1 ‘ ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) |
11 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
12 |
4 3 1 5 6 7 8
|
coe1tm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 0 ∈ ℕ0 ) → ( coe1 ‘ ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 0 , 𝑋 , 0 ) ) ) |
13 |
11 12
|
mp3an3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( coe1 ‘ ( 𝑋 ( ·𝑠 ‘ 𝑃 ) ( 0 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 0 , 𝑋 , 0 ) ) ) |
14 |
10 13
|
eqtrd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( coe1 ‘ ( 𝐴 ‘ 𝑋 ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 0 , 𝑋 , 0 ) ) ) |