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 |
|
coe1tmmul.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
9 |
|
coe1tmmul.t |
⊢ ∙ = ( .r ‘ 𝑃 ) |
10 |
|
coe1tmmul.u |
⊢ × = ( .r ‘ 𝑅 ) |
11 |
|
coe1tmmul.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) |
12 |
|
coe1tmmul.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
13 |
|
coe1tmmul.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐾 ) |
14 |
|
coe1tmmul.d |
⊢ ( 𝜑 → 𝐷 ∈ ℕ0 ) |
15 |
|
coe1tmmul2fv.y |
⊢ ( 𝜑 → 𝑌 ∈ ℕ0 ) |
16 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
coe1tmmul2 |
⊢ ( 𝜑 → ( coe1 ‘ ( 𝐴 ∙ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ) |
17 |
16
|
fveq1d |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐴 ∙ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ) ‘ ( 𝐷 + 𝑌 ) ) = ( ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ‘ ( 𝐷 + 𝑌 ) ) ) |
18 |
14 15
|
nn0addcld |
⊢ ( 𝜑 → ( 𝐷 + 𝑌 ) ∈ ℕ0 ) |
19 |
|
breq2 |
⊢ ( 𝑥 = ( 𝐷 + 𝑌 ) → ( 𝐷 ≤ 𝑥 ↔ 𝐷 ≤ ( 𝐷 + 𝑌 ) ) ) |
20 |
|
fvoveq1 |
⊢ ( 𝑥 = ( 𝐷 + 𝑌 ) → ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) = ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) ) |
21 |
20
|
oveq1d |
⊢ ( 𝑥 = ( 𝐷 + 𝑌 ) → ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) ) |
22 |
19 21
|
ifbieq1d |
⊢ ( 𝑥 = ( 𝐷 + 𝑌 ) → if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) = if ( 𝐷 ≤ ( 𝐷 + 𝑌 ) , ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) , 0 ) ) |
23 |
|
eqid |
⊢ ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) |
24 |
|
ovex |
⊢ ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) ∈ V |
25 |
1
|
fvexi |
⊢ 0 ∈ V |
26 |
24 25
|
ifex |
⊢ if ( 𝐷 ≤ ( 𝐷 + 𝑌 ) , ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) , 0 ) ∈ V |
27 |
22 23 26
|
fvmpt |
⊢ ( ( 𝐷 + 𝑌 ) ∈ ℕ0 → ( ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ‘ ( 𝐷 + 𝑌 ) ) = if ( 𝐷 ≤ ( 𝐷 + 𝑌 ) , ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) , 0 ) ) |
28 |
18 27
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ‘ ( 𝐷 + 𝑌 ) ) = if ( 𝐷 ≤ ( 𝐷 + 𝑌 ) , ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) , 0 ) ) |
29 |
14
|
nn0red |
⊢ ( 𝜑 → 𝐷 ∈ ℝ ) |
30 |
|
nn0addge1 |
⊢ ( ( 𝐷 ∈ ℝ ∧ 𝑌 ∈ ℕ0 ) → 𝐷 ≤ ( 𝐷 + 𝑌 ) ) |
31 |
29 15 30
|
syl2anc |
⊢ ( 𝜑 → 𝐷 ≤ ( 𝐷 + 𝑌 ) ) |
32 |
31
|
iftrued |
⊢ ( 𝜑 → if ( 𝐷 ≤ ( 𝐷 + 𝑌 ) , ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) , 0 ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) ) |
33 |
14
|
nn0cnd |
⊢ ( 𝜑 → 𝐷 ∈ ℂ ) |
34 |
15
|
nn0cnd |
⊢ ( 𝜑 → 𝑌 ∈ ℂ ) |
35 |
33 34
|
pncan2d |
⊢ ( 𝜑 → ( ( 𝐷 + 𝑌 ) − 𝐷 ) = 𝑌 ) |
36 |
35
|
fveq2d |
⊢ ( 𝜑 → ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) = ( ( coe1 ‘ 𝐴 ) ‘ 𝑌 ) ) |
37 |
36
|
oveq1d |
⊢ ( 𝜑 → ( ( ( coe1 ‘ 𝐴 ) ‘ ( ( 𝐷 + 𝑌 ) − 𝐷 ) ) × 𝐶 ) = ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑌 ) × 𝐶 ) ) |
38 |
28 32 37
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ‘ ( 𝐷 + 𝑌 ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑌 ) × 𝐶 ) ) |
39 |
17 38
|
eqtrd |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐴 ∙ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ) ‘ ( 𝐷 + 𝑌 ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑌 ) × 𝐶 ) ) |