Step |
Hyp |
Ref |
Expression |
1 |
|
deg1z.d |
⊢ 𝐷 = ( deg1 ‘ 𝑅 ) |
2 |
|
deg1z.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
deg1z.z |
⊢ 0 = ( 0g ‘ 𝑃 ) |
4 |
|
deg1nn0cl.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
5 |
|
deg1ldg.y |
⊢ 𝑌 = ( 0g ‘ 𝑅 ) |
6 |
|
deg1ldg.a |
⊢ 𝐴 = ( coe1 ‘ 𝐹 ) |
7 |
|
deg1ldgn.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
8 |
|
deg1ldgn.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
9 |
|
deg1ldgn.x |
⊢ ( 𝜑 → 𝑋 ∈ ℕ0 ) |
10 |
|
deg1ldgn.e |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑋 ) = 𝑌 ) |
11 |
|
fveq2 |
⊢ ( ( 𝐷 ‘ 𝐹 ) = 𝑋 → ( 𝐴 ‘ ( 𝐷 ‘ 𝐹 ) ) = ( 𝐴 ‘ 𝑋 ) ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝐷 ‘ 𝐹 ) = 𝑋 ) → ( 𝐴 ‘ ( 𝐷 ‘ 𝐹 ) ) = ( 𝐴 ‘ 𝑋 ) ) |
13 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐷 ‘ 𝐹 ) = 𝑋 ) → 𝑅 ∈ Ring ) |
14 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐷 ‘ 𝐹 ) = 𝑋 ) → 𝐹 ∈ 𝐵 ) |
15 |
|
eleq1a |
⊢ ( 𝑋 ∈ ℕ0 → ( ( 𝐷 ‘ 𝐹 ) = 𝑋 → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) ) |
16 |
9 15
|
syl |
⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝐹 ) = 𝑋 → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) ) |
17 |
16
|
imp |
⊢ ( ( 𝜑 ∧ ( 𝐷 ‘ 𝐹 ) = 𝑋 ) → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) |
18 |
1 2 3 4
|
deg1nn0clb |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ) → ( 𝐹 ≠ 0 ↔ ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) ) |
19 |
7 8 18
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ≠ 0 ↔ ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐷 ‘ 𝐹 ) = 𝑋 ) → ( 𝐹 ≠ 0 ↔ ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) ) |
21 |
17 20
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝐷 ‘ 𝐹 ) = 𝑋 ) → 𝐹 ≠ 0 ) |
22 |
1 2 3 4 5 6
|
deg1ldg |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ( 𝐴 ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ 𝑌 ) |
23 |
13 14 21 22
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝐷 ‘ 𝐹 ) = 𝑋 ) → ( 𝐴 ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ 𝑌 ) |
24 |
12 23
|
eqnetrrd |
⊢ ( ( 𝜑 ∧ ( 𝐷 ‘ 𝐹 ) = 𝑋 ) → ( 𝐴 ‘ 𝑋 ) ≠ 𝑌 ) |
25 |
24
|
ex |
⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝐹 ) = 𝑋 → ( 𝐴 ‘ 𝑋 ) ≠ 𝑌 ) ) |
26 |
25
|
necon2d |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑋 ) = 𝑌 → ( 𝐷 ‘ 𝐹 ) ≠ 𝑋 ) ) |
27 |
10 26
|
mpd |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ≠ 𝑋 ) |