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 |
2 3 4 5 6 7 8
|
ply1tmcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0 ) → ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∈ 𝐵 ) |
16 |
12 13 14 15
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∈ 𝐵 ) |
17 |
3 9 10 8
|
coe1mul |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∈ 𝐵 ) → ( coe1 ‘ ( 𝐴 ∙ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ) |
18 |
12 11 16 17
|
syl3anc |
⊢ ( 𝜑 → ( coe1 ‘ ( 𝐴 ∙ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ) |
19 |
|
eqeq2 |
⊢ ( ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) = if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) → ( ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) ↔ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ) |
20 |
|
eqeq2 |
⊢ ( 0 = if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) → ( ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = 0 ↔ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ) |
21 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝑅 ∈ Ring ) |
22 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
23 |
21 22
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝑅 ∈ Mnd ) |
24 |
|
ovex |
⊢ ( 0 ... 𝑥 ) ∈ V |
25 |
24
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 0 ... 𝑥 ) ∈ V ) |
26 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝐷 ≤ 𝑥 ) |
27 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝐷 ∈ ℕ0 ) |
28 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝑥 ∈ ℕ0 ) |
29 |
|
nn0sub |
⊢ ( ( 𝐷 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0 ) → ( 𝐷 ≤ 𝑥 ↔ ( 𝑥 − 𝐷 ) ∈ ℕ0 ) ) |
30 |
27 28 29
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 𝐷 ≤ 𝑥 ↔ ( 𝑥 − 𝐷 ) ∈ ℕ0 ) ) |
31 |
26 30
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 𝑥 − 𝐷 ) ∈ ℕ0 ) |
32 |
27
|
nn0ge0d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 0 ≤ 𝐷 ) |
33 |
|
nn0re |
⊢ ( 𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ ) |
34 |
33
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝑥 ∈ ℝ ) |
35 |
14
|
nn0red |
⊢ ( 𝜑 → 𝐷 ∈ ℝ ) |
36 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝐷 ∈ ℝ ) |
37 |
34 36
|
subge02d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 0 ≤ 𝐷 ↔ ( 𝑥 − 𝐷 ) ≤ 𝑥 ) ) |
38 |
32 37
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 𝑥 − 𝐷 ) ≤ 𝑥 ) |
39 |
|
fznn0 |
⊢ ( 𝑥 ∈ ℕ0 → ( ( 𝑥 − 𝐷 ) ∈ ( 0 ... 𝑥 ) ↔ ( ( 𝑥 − 𝐷 ) ∈ ℕ0 ∧ ( 𝑥 − 𝐷 ) ≤ 𝑥 ) ) ) |
40 |
39
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( ( 𝑥 − 𝐷 ) ∈ ( 0 ... 𝑥 ) ↔ ( ( 𝑥 − 𝐷 ) ∈ ℕ0 ∧ ( 𝑥 − 𝐷 ) ≤ 𝑥 ) ) ) |
41 |
31 38 40
|
mpbir2and |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 𝑥 − 𝐷 ) ∈ ( 0 ... 𝑥 ) ) |
42 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑅 ∈ Ring ) |
43 |
|
eqid |
⊢ ( coe1 ‘ 𝐴 ) = ( coe1 ‘ 𝐴 ) |
44 |
43 8 3 2
|
coe1f |
⊢ ( 𝐴 ∈ 𝐵 → ( coe1 ‘ 𝐴 ) : ℕ0 ⟶ 𝐾 ) |
45 |
11 44
|
syl |
⊢ ( 𝜑 → ( coe1 ‘ 𝐴 ) : ℕ0 ⟶ 𝐾 ) |
46 |
45
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( coe1 ‘ 𝐴 ) : ℕ0 ⟶ 𝐾 ) |
47 |
|
elfznn0 |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) → 𝑦 ∈ ℕ0 ) |
48 |
47
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑦 ∈ ℕ0 ) |
49 |
46 48
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) ∈ 𝐾 ) |
50 |
|
eqid |
⊢ ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) = ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) |
51 |
50 8 3 2
|
coe1f |
⊢ ( ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∈ 𝐵 → ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) : ℕ0 ⟶ 𝐾 ) |
52 |
16 51
|
syl |
⊢ ( 𝜑 → ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) : ℕ0 ⟶ 𝐾 ) |
53 |
52
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) : ℕ0 ⟶ 𝐾 ) |
54 |
|
fznn0sub |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) → ( 𝑥 − 𝑦 ) ∈ ℕ0 ) |
55 |
54
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( 𝑥 − 𝑦 ) ∈ ℕ0 ) |
56 |
53 55
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ∈ 𝐾 ) |
57 |
2 10
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) ∈ 𝐾 ∧ ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ∈ 𝐾 ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ∈ 𝐾 ) |
58 |
42 49 56 57
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ∈ 𝐾 ) |
59 |
58
|
fmpttd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) : ( 0 ... 𝑥 ) ⟶ 𝐾 ) |
60 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → 𝑅 ∈ Ring ) |
61 |
13
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → 𝐶 ∈ 𝐾 ) |
62 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → 𝐷 ∈ ℕ0 ) |
63 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) → 𝑦 ∈ ( 0 ... 𝑥 ) ) |
64 |
63 54
|
syl |
⊢ ( 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) → ( 𝑥 − 𝑦 ) ∈ ℕ0 ) |
65 |
64
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → ( 𝑥 − 𝑦 ) ∈ ℕ0 ) |
66 |
|
eldifsn |
⊢ ( 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ↔ ( 𝑦 ∈ ( 0 ... 𝑥 ) ∧ 𝑦 ≠ ( 𝑥 − 𝐷 ) ) ) |
67 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑥 ∈ ℕ0 ) |
68 |
67
|
nn0cnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑥 ∈ ℂ ) |
69 |
47
|
nn0cnd |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) → 𝑦 ∈ ℂ ) |
70 |
69
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑦 ∈ ℂ ) |
71 |
68 70
|
nncand |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( 𝑥 − ( 𝑥 − 𝑦 ) ) = 𝑦 ) |
72 |
71
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑦 = ( 𝑥 − ( 𝑥 − 𝑦 ) ) ) |
73 |
|
oveq2 |
⊢ ( 𝐷 = ( 𝑥 − 𝑦 ) → ( 𝑥 − 𝐷 ) = ( 𝑥 − ( 𝑥 − 𝑦 ) ) ) |
74 |
73
|
eqeq2d |
⊢ ( 𝐷 = ( 𝑥 − 𝑦 ) → ( 𝑦 = ( 𝑥 − 𝐷 ) ↔ 𝑦 = ( 𝑥 − ( 𝑥 − 𝑦 ) ) ) ) |
75 |
72 74
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( 𝐷 = ( 𝑥 − 𝑦 ) → 𝑦 = ( 𝑥 − 𝐷 ) ) ) |
76 |
75
|
necon3d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( 𝑦 ≠ ( 𝑥 − 𝐷 ) → 𝐷 ≠ ( 𝑥 − 𝑦 ) ) ) |
77 |
76
|
impr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ ( 𝑦 ∈ ( 0 ... 𝑥 ) ∧ 𝑦 ≠ ( 𝑥 − 𝐷 ) ) ) → 𝐷 ≠ ( 𝑥 − 𝑦 ) ) |
78 |
66 77
|
sylan2b |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → 𝐷 ≠ ( 𝑥 − 𝑦 ) ) |
79 |
1 2 3 4 5 6 7 60 61 62 65 78
|
coe1tmfv2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) = 0 ) |
80 |
79
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × 0 ) ) |
81 |
2 10 1
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) ∈ 𝐾 ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × 0 ) = 0 ) |
82 |
42 49 81
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × 0 ) = 0 ) |
83 |
63 82
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × 0 ) = 0 ) |
84 |
80 83
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { ( 𝑥 − 𝐷 ) } ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
85 |
84 25
|
suppss2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) supp 0 ) ⊆ { ( 𝑥 − 𝐷 ) } ) |
86 |
2 1 23 25 41 59 85
|
gsumpt |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ‘ ( 𝑥 − 𝐷 ) ) ) |
87 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑥 − 𝐷 ) → ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) = ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) |
88 |
|
oveq2 |
⊢ ( 𝑦 = ( 𝑥 − 𝐷 ) → ( 𝑥 − 𝑦 ) = ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) |
89 |
88
|
fveq2d |
⊢ ( 𝑦 = ( 𝑥 − 𝐷 ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) = ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) ) |
90 |
87 89
|
oveq12d |
⊢ ( 𝑦 = ( 𝑥 − 𝐷 ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) ) ) |
91 |
|
eqid |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) |
92 |
|
ovex |
⊢ ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) ) ∈ V |
93 |
90 91 92
|
fvmpt |
⊢ ( ( 𝑥 − 𝐷 ) ∈ ( 0 ... 𝑥 ) → ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ‘ ( 𝑥 − 𝐷 ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) ) ) |
94 |
41 93
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ‘ ( 𝑥 − 𝐷 ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) ) ) |
95 |
28
|
nn0cnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝑥 ∈ ℂ ) |
96 |
27
|
nn0cnd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝐷 ∈ ℂ ) |
97 |
95 96
|
nncand |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 𝑥 − ( 𝑥 − 𝐷 ) ) = 𝐷 ) |
98 |
97
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) = ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) ) |
99 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → 𝐶 ∈ 𝐾 ) |
100 |
1 2 3 4 5 6 7
|
coe1tmfv1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0 ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) = 𝐶 ) |
101 |
21 99 27 100
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) = 𝐶 ) |
102 |
98 101
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) = 𝐶 ) |
103 |
102
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − ( 𝑥 − 𝐷 ) ) ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) ) |
104 |
86 94 103
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) ) |
105 |
104
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) ) |
106 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝑅 ∈ Ring ) |
107 |
13
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝐶 ∈ 𝐾 ) |
108 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝐷 ∈ ℕ0 ) |
109 |
54
|
ad2antll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( 𝑥 − 𝑦 ) ∈ ℕ0 ) |
110 |
54
|
nn0red |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) → ( 𝑥 − 𝑦 ) ∈ ℝ ) |
111 |
110
|
ad2antll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( 𝑥 − 𝑦 ) ∈ ℝ ) |
112 |
33
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝑥 ∈ ℝ ) |
113 |
35
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝐷 ∈ ℝ ) |
114 |
47
|
ad2antll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝑦 ∈ ℕ0 ) |
115 |
114
|
nn0ge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 0 ≤ 𝑦 ) |
116 |
47
|
nn0red |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) → 𝑦 ∈ ℝ ) |
117 |
116
|
ad2antll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝑦 ∈ ℝ ) |
118 |
112 117
|
subge02d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( 0 ≤ 𝑦 ↔ ( 𝑥 − 𝑦 ) ≤ 𝑥 ) ) |
119 |
115 118
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( 𝑥 − 𝑦 ) ≤ 𝑥 ) |
120 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ¬ 𝐷 ≤ 𝑥 ) |
121 |
112 113
|
ltnled |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( 𝑥 < 𝐷 ↔ ¬ 𝐷 ≤ 𝑥 ) ) |
122 |
120 121
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝑥 < 𝐷 ) |
123 |
111 112 113 119 122
|
lelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( 𝑥 − 𝑦 ) < 𝐷 ) |
124 |
111 123
|
gtned |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → 𝐷 ≠ ( 𝑥 − 𝑦 ) ) |
125 |
1 2 3 4 5 6 7 106 107 108 109 124
|
coe1tmfv2 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) = 0 ) |
126 |
125
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) = ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × 0 ) ) |
127 |
45
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( coe1 ‘ 𝐴 ) : ℕ0 ⟶ 𝐾 ) |
128 |
127 114
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) ∈ 𝐾 ) |
129 |
106 128 81
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × 0 ) = 0 ) |
130 |
126 129
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ( ¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
131 |
130
|
anassrs |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
132 |
131
|
mpteq2dva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) |
133 |
132
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) ) |
134 |
12 22
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
135 |
1
|
gsumz |
⊢ ( ( 𝑅 ∈ Mnd ∧ ( 0 ... 𝑥 ) ∈ V ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) = 0 ) |
136 |
134 24 135
|
sylancl |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) = 0 ) |
137 |
136
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) = 0 ) |
138 |
133 137
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = 0 ) |
139 |
19 20 105 138
|
ifbothda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) |
140 |
139
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐴 ) ‘ 𝑦 ) × ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ) |
141 |
18 140
|
eqtrd |
⊢ ( 𝜑 → ( coe1 ‘ ( 𝐴 ∙ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) × 𝐶 ) , 0 ) ) ) |