Step |
Hyp |
Ref |
Expression |
1 |
|
ply1scl.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
ply1scl.a |
⊢ 𝐴 = ( algSc ‘ 𝑃 ) |
3 |
|
ply1scl0.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
ply1scl0.y |
⊢ 𝑌 = ( 0g ‘ 𝑃 ) |
5 |
|
ply1scln0.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
7 |
1 2 5 6
|
ply1sclf1 |
⊢ ( 𝑅 ∈ Ring → 𝐴 : 𝐾 –1-1→ ( Base ‘ 𝑃 ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → 𝐴 : 𝐾 –1-1→ ( Base ‘ 𝑃 ) ) |
9 |
|
simpr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → 𝑋 ∈ 𝐾 ) |
10 |
5 3
|
ring0cl |
⊢ ( 𝑅 ∈ Ring → 0 ∈ 𝐾 ) |
11 |
10
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → 0 ∈ 𝐾 ) |
12 |
|
f1fveq |
⊢ ( ( 𝐴 : 𝐾 –1-1→ ( Base ‘ 𝑃 ) ∧ ( 𝑋 ∈ 𝐾 ∧ 0 ∈ 𝐾 ) ) → ( ( 𝐴 ‘ 𝑋 ) = ( 𝐴 ‘ 0 ) ↔ 𝑋 = 0 ) ) |
13 |
8 9 11 12
|
syl12anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( ( 𝐴 ‘ 𝑋 ) = ( 𝐴 ‘ 0 ) ↔ 𝑋 = 0 ) ) |
14 |
13
|
biimpd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( ( 𝐴 ‘ 𝑋 ) = ( 𝐴 ‘ 0 ) → 𝑋 = 0 ) ) |
15 |
14
|
necon3d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ) → ( 𝑋 ≠ 0 → ( 𝐴 ‘ 𝑋 ) ≠ ( 𝐴 ‘ 0 ) ) ) |
16 |
15
|
3impia |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → ( 𝐴 ‘ 𝑋 ) ≠ ( 𝐴 ‘ 0 ) ) |
17 |
1 2 3 4
|
ply1scl0 |
⊢ ( 𝑅 ∈ Ring → ( 𝐴 ‘ 0 ) = 𝑌 ) |
18 |
17
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → ( 𝐴 ‘ 0 ) = 𝑌 ) |
19 |
16 18
|
neeqtrd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → ( 𝐴 ‘ 𝑋 ) ≠ 𝑌 ) |