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 ∧ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ) → ( coe1 ‘ ( ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∙ 𝐴 ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ) |
18 |
12 16 11 17
|
syl3anc |
⊢ ( 𝜑 → ( coe1 ‘ ( ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∙ 𝐴 ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ) |
19 |
|
eqeq2 |
⊢ ( ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) = if ( 𝐷 ≤ 𝑥 , ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) , 0 ) → ( ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) ↔ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = if ( 𝐷 ≤ 𝑥 , ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) , 0 ) ) ) |
20 |
|
eqeq2 |
⊢ ( 0 = if ( 𝐷 ≤ 𝑥 , ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) , 0 ) → ( ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = 0 ↔ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = if ( 𝐷 ≤ 𝑥 , ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) , 0 ) ) ) |
21 |
12
|
ad2antrr |
⊢ ( ( ( 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 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( 0 ... 𝑥 ) ∈ V ) |
25 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → 𝐷 ∈ ℕ0 ) |
26 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → 𝐷 ≤ 𝑥 ) |
27 |
|
fznn0 |
⊢ ( 𝑥 ∈ ℕ0 → ( 𝐷 ∈ ( 0 ... 𝑥 ) ↔ ( 𝐷 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ) |
28 |
27
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( 𝐷 ∈ ( 0 ... 𝑥 ) ↔ ( 𝐷 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥 ) ) ) |
29 |
25 26 28
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → 𝐷 ∈ ( 0 ... 𝑥 ) ) |
30 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑅 ∈ Ring ) |
31 |
|
eqid |
⊢ ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) = ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) |
32 |
31 8 3 2
|
coe1f |
⊢ ( ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∈ 𝐵 → ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) : ℕ0 ⟶ 𝐾 ) |
33 |
16 32
|
syl |
⊢ ( 𝜑 → ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) : ℕ0 ⟶ 𝐾 ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) → ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) : ℕ0 ⟶ 𝐾 ) |
35 |
|
elfznn0 |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) → 𝑦 ∈ ℕ0 ) |
36 |
|
ffvelrn |
⊢ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) : ℕ0 ⟶ 𝐾 ∧ 𝑦 ∈ ℕ0 ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) ∈ 𝐾 ) |
37 |
34 35 36
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) ∈ 𝐾 ) |
38 |
|
eqid |
⊢ ( coe1 ‘ 𝐴 ) = ( coe1 ‘ 𝐴 ) |
39 |
38 8 3 2
|
coe1f |
⊢ ( 𝐴 ∈ 𝐵 → ( coe1 ‘ 𝐴 ) : ℕ0 ⟶ 𝐾 ) |
40 |
11 39
|
syl |
⊢ ( 𝜑 → ( coe1 ‘ 𝐴 ) : ℕ0 ⟶ 𝐾 ) |
41 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) → ( coe1 ‘ 𝐴 ) : ℕ0 ⟶ 𝐾 ) |
42 |
|
fznn0sub |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) → ( 𝑥 − 𝑦 ) ∈ ℕ0 ) |
43 |
|
ffvelrn |
⊢ ( ( ( coe1 ‘ 𝐴 ) : ℕ0 ⟶ 𝐾 ∧ ( 𝑥 − 𝑦 ) ∈ ℕ0 ) → ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ∈ 𝐾 ) |
44 |
41 42 43
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ∈ 𝐾 ) |
45 |
2 10
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) ∈ 𝐾 ∧ ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ∈ 𝐾 ) → ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ∈ 𝐾 ) |
46 |
30 37 44 45
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ∈ 𝐾 ) |
47 |
46
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) → ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) : ( 0 ... 𝑥 ) ⟶ 𝐾 ) |
48 |
47
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) : ( 0 ... 𝑥 ) ⟶ 𝐾 ) |
49 |
12
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → 𝑅 ∈ Ring ) |
50 |
13
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → 𝐶 ∈ 𝐾 ) |
51 |
14
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → 𝐷 ∈ ℕ0 ) |
52 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) → 𝑦 ∈ ( 0 ... 𝑥 ) ) |
53 |
52 35
|
syl |
⊢ ( 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) → 𝑦 ∈ ℕ0 ) |
54 |
53
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → 𝑦 ∈ ℕ0 ) |
55 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) → 𝑦 ≠ 𝐷 ) |
56 |
55
|
necomd |
⊢ ( 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) → 𝐷 ≠ 𝑦 ) |
57 |
56
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → 𝐷 ≠ 𝑦 ) |
58 |
1 2 3 4 5 6 7 49 50 51 54 57
|
coe1tmfv2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) = 0 ) |
59 |
58
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = ( 0 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) |
60 |
2 10 1
|
ringlz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ∈ 𝐾 ) → ( 0 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
61 |
30 44 60
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( 0 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
62 |
52 61
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → ( 0 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
63 |
62
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → ( 0 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
64 |
59 63
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( ( 0 ... 𝑥 ) ∖ { 𝐷 } ) ) → ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
65 |
64 24
|
suppss2 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) supp 0 ) ⊆ { 𝐷 } ) |
66 |
2 1 23 24 29 48 65
|
gsumpt |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ‘ 𝐷 ) ) |
67 |
|
fveq2 |
⊢ ( 𝑦 = 𝐷 → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) = ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) ) |
68 |
|
oveq2 |
⊢ ( 𝑦 = 𝐷 → ( 𝑥 − 𝑦 ) = ( 𝑥 − 𝐷 ) ) |
69 |
68
|
fveq2d |
⊢ ( 𝑦 = 𝐷 → ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) = ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) |
70 |
67 69
|
oveq12d |
⊢ ( 𝑦 = 𝐷 → ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) ) |
71 |
|
eqid |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) |
72 |
|
ovex |
⊢ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) ∈ V |
73 |
70 71 72
|
fvmpt |
⊢ ( 𝐷 ∈ ( 0 ... 𝑥 ) → ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ‘ 𝐷 ) = ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) ) |
74 |
29 73
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ‘ 𝐷 ) = ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) ) |
75 |
1 2 3 4 5 6 7
|
coe1tmfv1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0 ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) = 𝐶 ) |
76 |
12 13 14 75
|
syl3anc |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) = 𝐶 ) |
77 |
76
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) = 𝐶 ) |
78 |
77
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝐷 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) = ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) ) |
79 |
74 78
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ‘ 𝐷 ) = ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) ) |
80 |
66 79
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) ) |
81 |
12
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑅 ∈ Ring ) |
82 |
13
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝐶 ∈ 𝐾 ) |
83 |
14
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝐷 ∈ ℕ0 ) |
84 |
35
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑦 ∈ ℕ0 ) |
85 |
|
elfzle2 |
⊢ ( 𝑦 ∈ ( 0 ... 𝑥 ) → 𝑦 ≤ 𝑥 ) |
86 |
85
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝑦 ≤ 𝑥 ) |
87 |
|
breq1 |
⊢ ( 𝐷 = 𝑦 → ( 𝐷 ≤ 𝑥 ↔ 𝑦 ≤ 𝑥 ) ) |
88 |
86 87
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( 𝐷 = 𝑦 → 𝐷 ≤ 𝑥 ) ) |
89 |
88
|
necon3bd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ¬ 𝐷 ≤ 𝑥 → 𝐷 ≠ 𝑦 ) ) |
90 |
89
|
imp |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) ∧ ¬ 𝐷 ≤ 𝑥 ) → 𝐷 ≠ 𝑦 ) |
91 |
90
|
an32s |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → 𝐷 ≠ 𝑦 ) |
92 |
1 2 3 4 5 6 7 81 82 83 84 91
|
coe1tmfv2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) = 0 ) |
93 |
92
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = ( 0 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) |
94 |
61
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( 0 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
95 |
93 94
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) ∧ 𝑦 ∈ ( 0 ... 𝑥 ) ) → ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) = 0 ) |
96 |
95
|
mpteq2dva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) |
97 |
96
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) ) |
98 |
12 22
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
99 |
98
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → 𝑅 ∈ Mnd ) |
100 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 0 ... 𝑥 ) ∈ V ) |
101 |
1
|
gsumz |
⊢ ( ( 𝑅 ∈ Mnd ∧ ( 0 ... 𝑥 ) ∈ V ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) = 0 ) |
102 |
99 100 101
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ 0 ) ) = 0 ) |
103 |
97 102
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) ∧ ¬ 𝐷 ≤ 𝑥 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = 0 ) |
104 |
19 20 80 103
|
ifbothda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℕ0 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = if ( 𝐷 ≤ 𝑥 , ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) , 0 ) ) |
105 |
104
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ) ‘ 𝑦 ) × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) , 0 ) ) ) |
106 |
18 105
|
eqtrd |
⊢ ( 𝜑 → ( coe1 ‘ ( ( 𝐶 · ( 𝐷 ↑ 𝑋 ) ) ∙ 𝐴 ) ) = ( 𝑥 ∈ ℕ0 ↦ if ( 𝐷 ≤ 𝑥 , ( 𝐶 × ( ( coe1 ‘ 𝐴 ) ‘ ( 𝑥 − 𝐷 ) ) ) , 0 ) ) ) |