Step |
Hyp |
Ref |
Expression |
1 |
|
coe1tm.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
2 |
|
coe1tm.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
3 |
|
coe1tm.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
4 |
|
coe1tm.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
5 |
|
coe1tm.m |
⊢ · = ( ·𝑠 ‘ 𝑃 ) |
6 |
|
coe1tm.n |
⊢ 𝑁 = ( mulGrp ‘ 𝑃 ) |
7 |
|
coe1tm.e |
⊢ ↑ = ( .g ‘ 𝑁 ) |
8 |
|
coe1tmfv2.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
9 |
|
coe1tmfv2.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐾 ) |
10 |
|
coe1tmfv2.d |
⊢ ( 𝜑 → 𝐷 ∈ ℕ0 ) |
11 |
|
coe1tmfv2.f |
⊢ ( 𝜑 → 𝐹 ∈ ℕ0 ) |
12 |
|
coe1tmfv2.q |
⊢ ( 𝜑 → 𝐷 ≠ 𝐹 ) |
13 |
1 2 3 4 5 6 7
|
coe1tm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0 ) → ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 𝐷 , 𝐶 , 0 ) ) ) |
14 |
8 9 10 13
|
syl3anc |
⊢ ( 𝜑 → ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 𝐷 , 𝐶 , 0 ) ) ) |
15 |
14
|
fveq1d |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐹 ) = ( ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 𝐷 , 𝐶 , 0 ) ) ‘ 𝐹 ) ) |
16 |
|
eqid |
⊢ ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 𝐷 , 𝐶 , 0 ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 𝐷 , 𝐶 , 0 ) ) |
17 |
|
eqeq1 |
⊢ ( 𝑥 = 𝐹 → ( 𝑥 = 𝐷 ↔ 𝐹 = 𝐷 ) ) |
18 |
17
|
ifbid |
⊢ ( 𝑥 = 𝐹 → if ( 𝑥 = 𝐷 , 𝐶 , 0 ) = if ( 𝐹 = 𝐷 , 𝐶 , 0 ) ) |
19 |
2 1
|
ring0cl |
⊢ ( 𝑅 ∈ Ring → 0 ∈ 𝐾 ) |
20 |
8 19
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝐾 ) |
21 |
9 20
|
ifcld |
⊢ ( 𝜑 → if ( 𝐹 = 𝐷 , 𝐶 , 0 ) ∈ 𝐾 ) |
22 |
16 18 11 21
|
fvmptd3 |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℕ0 ↦ if ( 𝑥 = 𝐷 , 𝐶 , 0 ) ) ‘ 𝐹 ) = if ( 𝐹 = 𝐷 , 𝐶 , 0 ) ) |
23 |
12
|
necomd |
⊢ ( 𝜑 → 𝐹 ≠ 𝐷 ) |
24 |
23
|
neneqd |
⊢ ( 𝜑 → ¬ 𝐹 = 𝐷 ) |
25 |
24
|
iffalsed |
⊢ ( 𝜑 → if ( 𝐹 = 𝐷 , 𝐶 , 0 ) = 0 ) |
26 |
15 22 25
|
3eqtrd |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐹 ) = 0 ) |