Step | Hyp | Ref
| Expression |
1 | | cply1coe0.k |
. . . . . 6
⊢ 𝐾 = (Base‘𝑅) |
2 | | cply1coe0.0 |
. . . . . 6
⊢ 0 =
(0g‘𝑅) |
3 | | cply1coe0.p |
. . . . . 6
⊢ 𝑃 = (Poly1‘𝑅) |
4 | | cply1coe0.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑃) |
5 | | cply1coe0.a |
. . . . . 6
⊢ 𝐴 = (algSc‘𝑃) |
6 | 1, 2, 3, 4, 5 | cply1coe0 21470 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝐾) → ∀𝑛 ∈ ℕ
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ) |
7 | 6 | ad4ant13 748 |
. . . 4
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ) |
8 | | fveq2 6774 |
. . . . . . . 8
⊢ (𝑀 = (𝐴‘𝑠) → (coe1‘𝑀) =
(coe1‘(𝐴‘𝑠))) |
9 | 8 | fveq1d 6776 |
. . . . . . 7
⊢ (𝑀 = (𝐴‘𝑠) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘𝑠))‘𝑛)) |
10 | 9 | eqeq1d 2740 |
. . . . . 6
⊢ (𝑀 = (𝐴‘𝑠) → (((coe1‘𝑀)‘𝑛) = 0 ↔
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )) |
11 | 10 | ralbidv 3112 |
. . . . 5
⊢ (𝑀 = (𝐴‘𝑠) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )) |
12 | 11 | adantl 482 |
. . . 4
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )) |
13 | 7, 12 | mpbird 256 |
. . 3
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) |
14 | 13 | rexlimdva2 3216 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 )) |
15 | | simpr 485 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) |
16 | | 0nn0 12248 |
. . . . . 6
⊢ 0 ∈
ℕ0 |
17 | | eqid 2738 |
. . . . . . 7
⊢
(coe1‘𝑀) = (coe1‘𝑀) |
18 | 17, 4, 3, 1 | coe1fvalcl 21383 |
. . . . . 6
⊢ ((𝑀 ∈ 𝐵 ∧ 0 ∈ ℕ0) →
((coe1‘𝑀)‘0) ∈ 𝐾) |
19 | 15, 16, 18 | sylancl 586 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ 𝐾) |
20 | 19 | adantr 481 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) →
((coe1‘𝑀)‘0) ∈ 𝐾) |
21 | | fveq2 6774 |
. . . . . 6
⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝐴‘𝑠) = (𝐴‘((coe1‘𝑀)‘0))) |
22 | 21 | eqeq2d 2749 |
. . . . 5
⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝑀 = (𝐴‘𝑠) ↔ 𝑀 = (𝐴‘((coe1‘𝑀)‘0)))) |
23 | 22 | adantl 482 |
. . . 4
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) ∧ 𝑠 = ((coe1‘𝑀)‘0)) → (𝑀 = (𝐴‘𝑠) ↔ 𝑀 = (𝐴‘((coe1‘𝑀)‘0)))) |
24 | | simpl 483 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
25 | | eqid 2738 |
. . . . . . . . . 10
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
26 | 3 | ply1ring 21419 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
27 | 3 | ply1lmod 21423 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
28 | | eqid 2738 |
. . . . . . . . . 10
⊢
(Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) |
29 | 5, 25, 26, 27, 28, 4 | asclf 21086 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵) |
30 | 29 | adantr 481 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵) |
31 | | eqid 2738 |
. . . . . . . . . . 11
⊢
(Base‘𝑅) =
(Base‘𝑅) |
32 | 17, 4, 3, 31 | coe1fvalcl 21383 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ 𝐵 ∧ 0 ∈ ℕ0) →
((coe1‘𝑀)‘0) ∈ (Base‘𝑅)) |
33 | 15, 16, 32 | sylancl 586 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈
(Base‘𝑅)) |
34 | 3 | ply1sca 21424 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
35 | 34 | eqcomd 2744 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ Ring →
(Scalar‘𝑃) = 𝑅) |
36 | 35 | fveq2d 6778 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring →
(Base‘(Scalar‘𝑃)) = (Base‘𝑅)) |
37 | 36 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅)) |
38 | 33, 37 | eleqtrrd 2842 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈
(Base‘(Scalar‘𝑃))) |
39 | 30, 38 | ffvelrnd 6962 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵) |
40 | 24, 15, 39 | 3jca 1127 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵)) |
41 | 40 | adantr 481 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵)) |
42 | | simpr 485 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧
((coe1‘𝑀)‘𝑛) = 0 ) →
((coe1‘𝑀)‘𝑛) = 0 ) |
43 | 3, 5, 1, 2 | coe1scl 21458 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧
((coe1‘𝑀)‘0) ∈ 𝐾) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0
↦ if(𝑘 = 0,
((coe1‘𝑀)‘0), 0 ))) |
44 | 19, 43 | syldan 591 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0
↦ if(𝑘 = 0,
((coe1‘𝑀)‘0), 0 ))) |
45 | 44 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) →
(coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0
↦ if(𝑘 = 0,
((coe1‘𝑀)‘0), 0 ))) |
46 | | nnne0 12007 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) |
47 | 46 | neneqd 2948 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → ¬
𝑛 = 0) |
48 | 47 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0) |
49 | 48 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑛 = 0) |
50 | | eqeq1 2742 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0)) |
51 | 50 | notbid 318 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0)) |
52 | 51 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0)) |
53 | 49, 52 | mpbird 256 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑘 = 0) |
54 | 53 | iffalsed 4470 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 ) = 0 ) |
55 | | nnnn0 12240 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
56 | 55 | adantl 482 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
57 | 2 | fvexi 6788 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
58 | 57 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 ∈ V) |
59 | 45, 54, 56, 58 | fvmptd 6882 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) →
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = 0 ) |
60 | 59 | eqcomd 2744 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
61 | 60 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧
((coe1‘𝑀)‘𝑛) = 0 ) → 0 =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
62 | 42, 61 | eqtrd 2778 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧
((coe1‘𝑀)‘𝑛) = 0 ) →
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
63 | 62 | ex 413 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) →
(((coe1‘𝑀)‘𝑛) = 0 →
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))) |
64 | 63 | ralimdva 3108 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 → ∀𝑛 ∈ ℕ
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))) |
65 | 64 | imp 407 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
66 | 3, 5, 1 | ply1sclid 21459 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧
((coe1‘𝑀)‘0) ∈ 𝐾) → ((coe1‘𝑀)‘0) =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)) |
67 | 19, 66 | syldan 591 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)) |
68 | 67 | adantr 481 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) →
((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)) |
69 | | df-n0 12234 |
. . . . . . . 8
⊢
ℕ0 = (ℕ ∪ {0}) |
70 | 69 | raleqi 3346 |
. . . . . . 7
⊢
(∀𝑛 ∈
ℕ0 ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ ∀𝑛 ∈ (ℕ ∪
{0})((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
71 | | c0ex 10969 |
. . . . . . . 8
⊢ 0 ∈
V |
72 | | fveq2 6774 |
. . . . . . . . . 10
⊢ (𝑛 = 0 →
((coe1‘𝑀)‘𝑛) = ((coe1‘𝑀)‘0)) |
73 | | fveq2 6774 |
. . . . . . . . . 10
⊢ (𝑛 = 0 →
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)) |
74 | 72, 73 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑛 = 0 →
(((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔
((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))) |
75 | 74 | ralunsn 4825 |
. . . . . . . 8
⊢ (0 ∈
V → (∀𝑛 ∈
(ℕ ∪ {0})((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ∧
((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)))) |
76 | 71, 75 | mp1i 13 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (∀𝑛 ∈ (ℕ ∪
{0})((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ∧
((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)))) |
77 | 70, 76 | bitrid 282 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (∀𝑛 ∈ ℕ0
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ∧
((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)))) |
78 | 65, 68, 77 | mpbir2and 710 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ0
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
79 | | eqid 2738 |
. . . . . 6
⊢
(coe1‘(𝐴‘((coe1‘𝑀)‘0))) =
(coe1‘(𝐴‘((coe1‘𝑀)‘0))) |
80 | 3, 4, 17, 79 | eqcoe1ply1eq 21468 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵) → (∀𝑛 ∈ ℕ0
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) → 𝑀 = (𝐴‘((coe1‘𝑀)‘0)))) |
81 | 41, 78, 80 | sylc 65 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → 𝑀 = (𝐴‘((coe1‘𝑀)‘0))) |
82 | 20, 23, 81 | rspcedvd 3563 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠)) |
83 | 82 | ex 413 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠))) |
84 | 14, 83 | impbid 211 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 )) |