Step |
Hyp |
Ref |
Expression |
1 |
|
coe1mul3.s |
⊢ 𝑌 = ( Poly1 ‘ 𝑅 ) |
2 |
|
coe1mul3.t |
⊢ ∙ = ( .r ‘ 𝑌 ) |
3 |
|
coe1mul3.u |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
coe1mul3.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
5 |
|
coe1mul3.d |
⊢ 𝐷 = ( deg1 ‘ 𝑅 ) |
6 |
|
coe1mul3.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
7 |
|
coe1mul3.f1 |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
8 |
|
coe1mul3.f2 |
⊢ ( 𝜑 → 𝐼 ∈ ℕ0 ) |
9 |
|
coe1mul3.f3 |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ≤ 𝐼 ) |
10 |
|
coe1mul3.g1 |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
11 |
|
coe1mul3.g2 |
⊢ ( 𝜑 → 𝐽 ∈ ℕ0 ) |
12 |
|
coe1mul3.g3 |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) ≤ 𝐽 ) |
13 |
1 2 3 4
|
coe1mul |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( coe1 ‘ ( 𝐹 ∙ 𝐺 ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ) |
14 |
6 7 10 13
|
syl3anc |
⊢ ( 𝜑 → ( coe1 ‘ ( 𝐹 ∙ 𝐺 ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ) |
15 |
14
|
fveq1d |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐹 ∙ 𝐺 ) ) ‘ ( 𝐼 + 𝐽 ) ) = ( ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ‘ ( 𝐼 + 𝐽 ) ) ) |
16 |
8 11
|
nn0addcld |
⊢ ( 𝜑 → ( 𝐼 + 𝐽 ) ∈ ℕ0 ) |
17 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝐼 + 𝐽 ) → ( 0 ... 𝑥 ) = ( 0 ... ( 𝐼 + 𝐽 ) ) ) |
18 |
|
fvoveq1 |
⊢ ( 𝑥 = ( 𝐼 + 𝐽 ) → ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) = ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) |
19 |
18
|
oveq2d |
⊢ ( 𝑥 = ( 𝐼 + 𝐽 ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) |
20 |
17 19
|
mpteq12dv |
⊢ ( 𝑥 = ( 𝐼 + 𝐽 ) → ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ) |
21 |
20
|
oveq2d |
⊢ ( 𝑥 = ( 𝐼 + 𝐽 ) → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) = ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ) ) |
22 |
|
eqid |
⊢ ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) = ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) |
23 |
|
ovex |
⊢ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ) ∈ V |
24 |
21 22 23
|
fvmpt |
⊢ ( ( 𝐼 + 𝐽 ) ∈ ℕ0 → ( ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ‘ ( 𝐼 + 𝐽 ) ) = ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ) ) |
25 |
16 24
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℕ0 ↦ ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... 𝑥 ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝑥 − 𝑦 ) ) ) ) ) ) ‘ ( 𝐼 + 𝐽 ) ) = ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ) ) |
26 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
27 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
28 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
29 |
6 28
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
30 |
|
ovexd |
⊢ ( 𝜑 → ( 0 ... ( 𝐼 + 𝐽 ) ) ∈ V ) |
31 |
8
|
nn0red |
⊢ ( 𝜑 → 𝐼 ∈ ℝ ) |
32 |
|
nn0addge1 |
⊢ ( ( 𝐼 ∈ ℝ ∧ 𝐽 ∈ ℕ0 ) → 𝐼 ≤ ( 𝐼 + 𝐽 ) ) |
33 |
31 11 32
|
syl2anc |
⊢ ( 𝜑 → 𝐼 ≤ ( 𝐼 + 𝐽 ) ) |
34 |
|
fznn0 |
⊢ ( ( 𝐼 + 𝐽 ) ∈ ℕ0 → ( 𝐼 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↔ ( 𝐼 ∈ ℕ0 ∧ 𝐼 ≤ ( 𝐼 + 𝐽 ) ) ) ) |
35 |
16 34
|
syl |
⊢ ( 𝜑 → ( 𝐼 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↔ ( 𝐼 ∈ ℕ0 ∧ 𝐼 ≤ ( 𝐼 + 𝐽 ) ) ) ) |
36 |
8 33 35
|
mpbir2and |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) |
37 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → 𝑅 ∈ Ring ) |
38 |
|
eqid |
⊢ ( coe1 ‘ 𝐹 ) = ( coe1 ‘ 𝐹 ) |
39 |
38 4 1 26
|
coe1f |
⊢ ( 𝐹 ∈ 𝐵 → ( coe1 ‘ 𝐹 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) ) |
40 |
7 39
|
syl |
⊢ ( 𝜑 → ( coe1 ‘ 𝐹 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) ) |
41 |
|
elfznn0 |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) → 𝑦 ∈ ℕ0 ) |
42 |
|
ffvelrn |
⊢ ( ( ( coe1 ‘ 𝐹 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) |
43 |
40 41 42
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) |
44 |
|
eqid |
⊢ ( coe1 ‘ 𝐺 ) = ( coe1 ‘ 𝐺 ) |
45 |
44 4 1 26
|
coe1f |
⊢ ( 𝐺 ∈ 𝐵 → ( coe1 ‘ 𝐺 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) ) |
46 |
10 45
|
syl |
⊢ ( 𝜑 → ( coe1 ‘ 𝐺 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) ) |
47 |
|
fznn0sub |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) → ( ( 𝐼 + 𝐽 ) − 𝑦 ) ∈ ℕ0 ) |
48 |
|
ffvelrn |
⊢ ( ( ( coe1 ‘ 𝐺 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) ∧ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ∈ ℕ0 ) → ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ∈ ( Base ‘ 𝑅 ) ) |
49 |
46 47 48
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ∈ ( Base ‘ 𝑅 ) ) |
50 |
26 3
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ∧ ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
51 |
37 43 49 50
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
52 |
51
|
fmpttd |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) : ( 0 ... ( 𝐼 + 𝐽 ) ) ⟶ ( Base ‘ 𝑅 ) ) |
53 |
|
eldifsn |
⊢ ( 𝑦 ∈ ( ( 0 ... ( 𝐼 + 𝐽 ) ) ∖ { 𝐼 } ) ↔ ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ∧ 𝑦 ≠ 𝐼 ) ) |
54 |
41
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → 𝑦 ∈ ℕ0 ) |
55 |
54
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → 𝑦 ∈ ℝ ) |
56 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → 𝐼 ∈ ℝ ) |
57 |
55 56
|
lttri2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( 𝑦 ≠ 𝐼 ↔ ( 𝑦 < 𝐼 ∨ 𝐼 < 𝑦 ) ) ) |
58 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → 𝐺 ∈ 𝐵 ) |
59 |
47
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( 𝐼 + 𝐽 ) − 𝑦 ) ∈ ℕ0 ) |
60 |
59
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( ( 𝐼 + 𝐽 ) − 𝑦 ) ∈ ℕ0 ) |
61 |
5 1 4
|
deg1xrcl |
⊢ ( 𝐺 ∈ 𝐵 → ( 𝐷 ‘ 𝐺 ) ∈ ℝ* ) |
62 |
10 61
|
syl |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) ∈ ℝ* ) |
63 |
62
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( 𝐷 ‘ 𝐺 ) ∈ ℝ* ) |
64 |
11
|
nn0red |
⊢ ( 𝜑 → 𝐽 ∈ ℝ ) |
65 |
64
|
rexrd |
⊢ ( 𝜑 → 𝐽 ∈ ℝ* ) |
66 |
65
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → 𝐽 ∈ ℝ* ) |
67 |
16
|
nn0red |
⊢ ( 𝜑 → ( 𝐼 + 𝐽 ) ∈ ℝ ) |
68 |
67
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( 𝐼 + 𝐽 ) ∈ ℝ ) |
69 |
68 55
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( 𝐼 + 𝐽 ) − 𝑦 ) ∈ ℝ ) |
70 |
69
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( 𝐼 + 𝐽 ) − 𝑦 ) ∈ ℝ* ) |
71 |
70
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( ( 𝐼 + 𝐽 ) − 𝑦 ) ∈ ℝ* ) |
72 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( 𝐷 ‘ 𝐺 ) ≤ 𝐽 ) |
73 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → 𝐽 ∈ ℝ ) |
74 |
55 56 73
|
ltadd1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( 𝑦 < 𝐼 ↔ ( 𝑦 + 𝐽 ) < ( 𝐼 + 𝐽 ) ) ) |
75 |
55 73 68
|
ltaddsub2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( 𝑦 + 𝐽 ) < ( 𝐼 + 𝐽 ) ↔ 𝐽 < ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) |
76 |
74 75
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( 𝑦 < 𝐼 ↔ 𝐽 < ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) |
77 |
76
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → 𝐽 < ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) |
78 |
63 66 71 72 77
|
xrlelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( 𝐷 ‘ 𝐺 ) < ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) |
79 |
5 1 4 27 44
|
deg1lt |
⊢ ( ( 𝐺 ∈ 𝐵 ∧ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐺 ) < ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) → ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) = ( 0g ‘ 𝑅 ) ) |
80 |
58 60 78 79
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) = ( 0g ‘ 𝑅 ) ) |
81 |
80
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( 0g ‘ 𝑅 ) ) ) |
82 |
26 3 27
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
83 |
37 43 82
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
84 |
83
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
85 |
81 84
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝑦 < 𝐼 ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) |
86 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → 𝐹 ∈ 𝐵 ) |
87 |
54
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → 𝑦 ∈ ℕ0 ) |
88 |
5 1 4
|
deg1xrcl |
⊢ ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* ) |
89 |
7 88
|
syl |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* ) |
90 |
89
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* ) |
91 |
31
|
rexrd |
⊢ ( 𝜑 → 𝐼 ∈ ℝ* ) |
92 |
91
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → 𝐼 ∈ ℝ* ) |
93 |
55
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → 𝑦 ∈ ℝ* ) |
94 |
93
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → 𝑦 ∈ ℝ* ) |
95 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → ( 𝐷 ‘ 𝐹 ) ≤ 𝐼 ) |
96 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → 𝐼 < 𝑦 ) |
97 |
90 92 94 95 96
|
xrlelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → ( 𝐷 ‘ 𝐹 ) < 𝑦 ) |
98 |
5 1 4 27 38
|
deg1lt |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝑦 ) → ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) = ( 0g ‘ 𝑅 ) ) |
99 |
86 87 97 98
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) = ( 0g ‘ 𝑅 ) ) |
100 |
99
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( ( 0g ‘ 𝑅 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) |
101 |
26 3 27
|
ringlz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 0g ‘ 𝑅 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) |
102 |
37 49 101
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( 0g ‘ 𝑅 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) |
103 |
102
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → ( ( 0g ‘ 𝑅 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) |
104 |
100 103
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ 𝐼 < 𝑦 ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) |
105 |
85 104
|
jaodan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) ∧ ( 𝑦 < 𝐼 ∨ 𝐼 < 𝑦 ) ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) |
106 |
105
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( ( 𝑦 < 𝐼 ∨ 𝐼 < 𝑦 ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) ) |
107 |
57 106
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ) → ( 𝑦 ≠ 𝐼 → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) ) |
108 |
107
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ∧ 𝑦 ≠ 𝐼 ) ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) |
109 |
53 108
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 0 ... ( 𝐼 + 𝐽 ) ) ∖ { 𝐼 } ) ) → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( 0g ‘ 𝑅 ) ) |
110 |
109 30
|
suppss2 |
⊢ ( 𝜑 → ( ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) supp ( 0g ‘ 𝑅 ) ) ⊆ { 𝐼 } ) |
111 |
26 27 29 30 36 52 110
|
gsumpt |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ) = ( ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ‘ 𝐼 ) ) |
112 |
|
fveq2 |
⊢ ( 𝑦 = 𝐼 → ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) = ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) ) |
113 |
|
oveq2 |
⊢ ( 𝑦 = 𝐼 → ( ( 𝐼 + 𝐽 ) − 𝑦 ) = ( ( 𝐼 + 𝐽 ) − 𝐼 ) ) |
114 |
113
|
fveq2d |
⊢ ( 𝑦 = 𝐼 → ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) = ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝐼 ) ) ) |
115 |
112 114
|
oveq12d |
⊢ ( 𝑦 = 𝐼 → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝐼 ) ) ) ) |
116 |
|
eqid |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) |
117 |
|
ovex |
⊢ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝐼 ) ) ) ∈ V |
118 |
115 116 117
|
fvmpt |
⊢ ( 𝐼 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) → ( ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ‘ 𝐼 ) = ( ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝐼 ) ) ) ) |
119 |
36 118
|
syl |
⊢ ( 𝜑 → ( ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ‘ 𝐼 ) = ( ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝐼 ) ) ) ) |
120 |
8
|
nn0cnd |
⊢ ( 𝜑 → 𝐼 ∈ ℂ ) |
121 |
11
|
nn0cnd |
⊢ ( 𝜑 → 𝐽 ∈ ℂ ) |
122 |
120 121
|
pncan2d |
⊢ ( 𝜑 → ( ( 𝐼 + 𝐽 ) − 𝐼 ) = 𝐽 ) |
123 |
122
|
fveq2d |
⊢ ( 𝜑 → ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝐼 ) ) = ( ( coe1 ‘ 𝐺 ) ‘ 𝐽 ) ) |
124 |
123
|
oveq2d |
⊢ ( 𝜑 → ( ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝐼 ) ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) · ( ( coe1 ‘ 𝐺 ) ‘ 𝐽 ) ) ) |
125 |
111 119 124
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑦 ∈ ( 0 ... ( 𝐼 + 𝐽 ) ) ↦ ( ( ( coe1 ‘ 𝐹 ) ‘ 𝑦 ) · ( ( coe1 ‘ 𝐺 ) ‘ ( ( 𝐼 + 𝐽 ) − 𝑦 ) ) ) ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) · ( ( coe1 ‘ 𝐺 ) ‘ 𝐽 ) ) ) |
126 |
15 25 125
|
3eqtrd |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐹 ∙ 𝐺 ) ) ‘ ( 𝐼 + 𝐽 ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ 𝐼 ) · ( ( coe1 ‘ 𝐺 ) ‘ 𝐽 ) ) ) |