Step |
Hyp |
Ref |
Expression |
1 |
|
uc1pdeg.d |
⊢ 𝐷 = ( deg1 ‘ 𝑅 ) |
2 |
|
uc1pdeg.c |
⊢ 𝐶 = ( Unic1p ‘ 𝑅 ) |
3 |
|
simpl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶 ) → 𝑅 ∈ Ring ) |
4 |
|
eqid |
⊢ ( Poly1 ‘ 𝑅 ) = ( Poly1 ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( Base ‘ ( Poly1 ‘ 𝑅 ) ) = ( Base ‘ ( Poly1 ‘ 𝑅 ) ) |
6 |
4 5 2
|
uc1pcl |
⊢ ( 𝐹 ∈ 𝐶 → 𝐹 ∈ ( Base ‘ ( Poly1 ‘ 𝑅 ) ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶 ) → 𝐹 ∈ ( Base ‘ ( Poly1 ‘ 𝑅 ) ) ) |
8 |
|
eqid |
⊢ ( 0g ‘ ( Poly1 ‘ 𝑅 ) ) = ( 0g ‘ ( Poly1 ‘ 𝑅 ) ) |
9 |
4 8 2
|
uc1pn0 |
⊢ ( 𝐹 ∈ 𝐶 → 𝐹 ≠ ( 0g ‘ ( Poly1 ‘ 𝑅 ) ) ) |
10 |
9
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶 ) → 𝐹 ≠ ( 0g ‘ ( Poly1 ‘ 𝑅 ) ) ) |
11 |
1 4 8 5
|
deg1nn0cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ ( Base ‘ ( Poly1 ‘ 𝑅 ) ) ∧ 𝐹 ≠ ( 0g ‘ ( Poly1 ‘ 𝑅 ) ) ) → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) |
12 |
3 7 10 11
|
syl3anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶 ) → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) |