Step |
Hyp |
Ref |
Expression |
1 |
|
gsumply1eq.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
gsumply1eq.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
3 |
|
gsumply1eq.e |
⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑃 ) ) |
4 |
|
gsumply1eq.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
5 |
|
gsumply1eq.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
6 |
|
gsumply1eq.m |
⊢ ∗ = ( ·𝑠 ‘ 𝑃 ) |
7 |
|
gsumply1eq.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
8 |
|
gsumply1eq.a |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ0 𝐴 ∈ 𝐾 ) |
9 |
|
gsumply1eq.f1 |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ0 ↦ 𝐴 ) finSupp 0 ) |
10 |
|
gsumply1eq.b |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ0 𝐵 ∈ 𝐾 ) |
11 |
|
gsumply1eq.f2 |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ0 ↦ 𝐵 ) finSupp 0 ) |
12 |
|
gsumply1eq.o |
⊢ ( 𝜑 → 𝑂 = ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( 𝐴 ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) ) |
13 |
|
gsumply1eq.q |
⊢ ( 𝜑 → 𝑄 = ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( 𝐵 ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
15 |
1 14 2 3 4 5 6 7 8 9
|
gsumsmonply1 |
⊢ ( 𝜑 → ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( 𝐴 ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) ∈ ( Base ‘ 𝑃 ) ) |
16 |
12 15
|
eqeltrd |
⊢ ( 𝜑 → 𝑂 ∈ ( Base ‘ 𝑃 ) ) |
17 |
1 14 2 3 4 5 6 7 10 11
|
gsumsmonply1 |
⊢ ( 𝜑 → ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( 𝐵 ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) ∈ ( Base ‘ 𝑃 ) ) |
18 |
13 17
|
eqeltrd |
⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝑃 ) ) |
19 |
|
eqid |
⊢ ( coe1 ‘ 𝑂 ) = ( coe1 ‘ 𝑂 ) |
20 |
|
eqid |
⊢ ( coe1 ‘ 𝑄 ) = ( coe1 ‘ 𝑄 ) |
21 |
1 14 19 20
|
ply1coe1eq |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑂 ∈ ( Base ‘ 𝑃 ) ∧ 𝑄 ∈ ( Base ‘ 𝑃 ) ) → ( ∀ 𝑘 ∈ ℕ0 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) = ( ( coe1 ‘ 𝑄 ) ‘ 𝑘 ) ↔ 𝑂 = 𝑄 ) ) |
22 |
21
|
bicomd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑂 ∈ ( Base ‘ 𝑃 ) ∧ 𝑄 ∈ ( Base ‘ 𝑃 ) ) → ( 𝑂 = 𝑄 ↔ ∀ 𝑘 ∈ ℕ0 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) = ( ( coe1 ‘ 𝑄 ) ‘ 𝑘 ) ) ) |
23 |
4 16 18 22
|
syl3anc |
⊢ ( 𝜑 → ( 𝑂 = 𝑄 ↔ ∀ 𝑘 ∈ ℕ0 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) = ( ( coe1 ‘ 𝑄 ) ‘ 𝑘 ) ) ) |
24 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑂 = ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( 𝐴 ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) ) |
25 |
|
nfcv |
⊢ Ⅎ 𝑙 ( 𝐴 ∗ ( 𝑘 ↑ 𝑋 ) ) |
26 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑙 / 𝑘 ⦌ 𝐴 |
27 |
|
nfcv |
⊢ Ⅎ 𝑘 ∗ |
28 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 𝑙 ↑ 𝑋 ) |
29 |
26 27 28
|
nfov |
⊢ Ⅎ 𝑘 ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) |
30 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑙 → 𝐴 = ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ) |
31 |
|
oveq1 |
⊢ ( 𝑘 = 𝑙 → ( 𝑘 ↑ 𝑋 ) = ( 𝑙 ↑ 𝑋 ) ) |
32 |
30 31
|
oveq12d |
⊢ ( 𝑘 = 𝑙 → ( 𝐴 ∗ ( 𝑘 ↑ 𝑋 ) ) = ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) ) |
33 |
25 29 32
|
cbvmpt |
⊢ ( 𝑘 ∈ ℕ0 ↦ ( 𝐴 ∗ ( 𝑘 ↑ 𝑋 ) ) ) = ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) ) |
34 |
33
|
oveq2i |
⊢ ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( 𝐴 ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) = ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) |
35 |
24 34
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑂 = ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) |
36 |
35
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( coe1 ‘ 𝑂 ) = ( coe1 ‘ ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) ) |
37 |
36
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) = ( ( coe1 ‘ ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) ‘ 𝑘 ) ) |
38 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑅 ∈ Ring ) |
39 |
|
nfv |
⊢ Ⅎ 𝑙 𝐴 ∈ 𝐾 |
40 |
26
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∈ 𝐾 |
41 |
30
|
eleq1d |
⊢ ( 𝑘 = 𝑙 → ( 𝐴 ∈ 𝐾 ↔ ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∈ 𝐾 ) ) |
42 |
39 40 41
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ ℕ0 𝐴 ∈ 𝐾 ↔ ∀ 𝑙 ∈ ℕ0 ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∈ 𝐾 ) |
43 |
8 42
|
sylib |
⊢ ( 𝜑 → ∀ 𝑙 ∈ ℕ0 ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∈ 𝐾 ) |
44 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ∀ 𝑙 ∈ ℕ0 ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∈ 𝐾 ) |
45 |
|
nfcv |
⊢ Ⅎ 𝑙 𝐴 |
46 |
45 26 30
|
cbvmpt |
⊢ ( 𝑘 ∈ ℕ0 ↦ 𝐴 ) = ( 𝑙 ∈ ℕ0 ↦ ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ) |
47 |
46 9
|
eqbrtrrid |
⊢ ( 𝜑 → ( 𝑙 ∈ ℕ0 ↦ ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ) finSupp 0 ) |
48 |
47
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑙 ∈ ℕ0 ↦ ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ) finSupp 0 ) |
49 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
50 |
1 14 2 3 38 5 6 7 44 48 49
|
gsummoncoe1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) ‘ 𝑘 ) = ⦋ 𝑘 / 𝑙 ⦌ ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ) |
51 |
|
csbcow |
⊢ ⦋ 𝑘 / 𝑙 ⦌ ⦋ 𝑙 / 𝑘 ⦌ 𝐴 = ⦋ 𝑘 / 𝑘 ⦌ 𝐴 |
52 |
|
csbid |
⊢ ⦋ 𝑘 / 𝑘 ⦌ 𝐴 = 𝐴 |
53 |
51 52
|
eqtri |
⊢ ⦋ 𝑘 / 𝑙 ⦌ ⦋ 𝑙 / 𝑘 ⦌ 𝐴 = 𝐴 |
54 |
50 53
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐴 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) ‘ 𝑘 ) = 𝐴 ) |
55 |
37 54
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) = 𝐴 ) |
56 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑄 = ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( 𝐵 ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) ) |
57 |
|
nfcv |
⊢ Ⅎ 𝑙 ( 𝐵 ∗ ( 𝑘 ↑ 𝑋 ) ) |
58 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑙 / 𝑘 ⦌ 𝐵 |
59 |
58 27 28
|
nfov |
⊢ Ⅎ 𝑘 ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) |
60 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑙 → 𝐵 = ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ) |
61 |
60 31
|
oveq12d |
⊢ ( 𝑘 = 𝑙 → ( 𝐵 ∗ ( 𝑘 ↑ 𝑋 ) ) = ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) |
62 |
57 59 61
|
cbvmpt |
⊢ ( 𝑘 ∈ ℕ0 ↦ ( 𝐵 ∗ ( 𝑘 ↑ 𝑋 ) ) ) = ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) |
63 |
62
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ℕ0 ↦ ( 𝐵 ∗ ( 𝑘 ↑ 𝑋 ) ) ) = ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) |
64 |
63
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( 𝐵 ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) = ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) |
65 |
56 64
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑄 = ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) |
66 |
65
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( coe1 ‘ 𝑄 ) = ( coe1 ‘ ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) ) |
67 |
66
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( coe1 ‘ 𝑄 ) ‘ 𝑘 ) = ( ( coe1 ‘ ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) ‘ 𝑘 ) ) |
68 |
|
nfv |
⊢ Ⅎ 𝑙 𝐵 ∈ 𝐾 |
69 |
58
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∈ 𝐾 |
70 |
60
|
eleq1d |
⊢ ( 𝑘 = 𝑙 → ( 𝐵 ∈ 𝐾 ↔ ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∈ 𝐾 ) ) |
71 |
68 69 70
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ ℕ0 𝐵 ∈ 𝐾 ↔ ∀ 𝑙 ∈ ℕ0 ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∈ 𝐾 ) |
72 |
10 71
|
sylib |
⊢ ( 𝜑 → ∀ 𝑙 ∈ ℕ0 ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∈ 𝐾 ) |
73 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ∀ 𝑙 ∈ ℕ0 ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∈ 𝐾 ) |
74 |
|
nfcv |
⊢ Ⅎ 𝑙 𝐵 |
75 |
74 58 60
|
cbvmpt |
⊢ ( 𝑘 ∈ ℕ0 ↦ 𝐵 ) = ( 𝑙 ∈ ℕ0 ↦ ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ) |
76 |
75 11
|
eqbrtrrid |
⊢ ( 𝜑 → ( 𝑙 ∈ ℕ0 ↦ ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ) finSupp 0 ) |
77 |
76
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑙 ∈ ℕ0 ↦ ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ) finSupp 0 ) |
78 |
1 14 2 3 38 5 6 7 73 77 49
|
gsummoncoe1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) ‘ 𝑘 ) = ⦋ 𝑘 / 𝑙 ⦌ ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ) |
79 |
|
csbcow |
⊢ ⦋ 𝑘 / 𝑙 ⦌ ⦋ 𝑙 / 𝑘 ⦌ 𝐵 = ⦋ 𝑘 / 𝑘 ⦌ 𝐵 |
80 |
|
csbid |
⊢ ⦋ 𝑘 / 𝑘 ⦌ 𝐵 = 𝐵 |
81 |
79 80
|
eqtri |
⊢ ⦋ 𝑘 / 𝑙 ⦌ ⦋ 𝑙 / 𝑘 ⦌ 𝐵 = 𝐵 |
82 |
78 81
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑙 ∈ ℕ0 ↦ ( ⦋ 𝑙 / 𝑘 ⦌ 𝐵 ∗ ( 𝑙 ↑ 𝑋 ) ) ) ) ) ‘ 𝑘 ) = 𝐵 ) |
83 |
67 82
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( coe1 ‘ 𝑄 ) ‘ 𝑘 ) = 𝐵 ) |
84 |
55 83
|
eqeq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) = ( ( coe1 ‘ 𝑄 ) ‘ 𝑘 ) ↔ 𝐴 = 𝐵 ) ) |
85 |
84
|
ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ ℕ0 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) = ( ( coe1 ‘ 𝑄 ) ‘ 𝑘 ) ↔ ∀ 𝑘 ∈ ℕ0 𝐴 = 𝐵 ) ) |
86 |
23 85
|
bitrd |
⊢ ( 𝜑 → ( 𝑂 = 𝑄 ↔ ∀ 𝑘 ∈ ℕ0 𝐴 = 𝐵 ) ) |