Step | Hyp | Ref
| Expression |
1 | | simpl 472 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
2 | 1 | anim1i 590 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) → (𝑅 ∈ Ring ∧ 𝑠 ∈ 𝐾)) |
3 | 2 | adantr 480 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → (𝑅 ∈ Ring ∧ 𝑠 ∈ 𝐾)) |
4 | | cply1coe0.k |
. . . . . . 7
⊢ 𝐾 = (Base‘𝑅) |
5 | | cply1coe0.0 |
. . . . . . 7
⊢ 0 =
(0g‘𝑅) |
6 | | cply1coe0.p |
. . . . . . 7
⊢ 𝑃 = (Poly1‘𝑅) |
7 | | cply1coe0.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝑃) |
8 | | cply1coe0.a |
. . . . . . 7
⊢ 𝐴 = (algSc‘𝑃) |
9 | 4, 5, 6, 7, 8 | cply1coe0 19490 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝐾) → ∀𝑛 ∈ ℕ
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ) |
10 | 3, 9 | syl 17 |
. . . . 5
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ) |
11 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑀 = (𝐴‘𝑠) → (coe1‘𝑀) =
(coe1‘(𝐴‘𝑠))) |
12 | 11 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝑀 = (𝐴‘𝑠) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘𝑠))‘𝑛)) |
13 | 12 | eqeq1d 2612 |
. . . . . . 7
⊢ (𝑀 = (𝐴‘𝑠) → (((coe1‘𝑀)‘𝑛) = 0 ↔
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )) |
14 | 13 | ralbidv 2969 |
. . . . . 6
⊢ (𝑀 = (𝐴‘𝑠) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )) |
15 | 14 | adantl 481 |
. . . . 5
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ
((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )) |
16 | 10, 15 | mpbird 246 |
. . . 4
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) |
17 | 16 | ex 449 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) → (𝑀 = (𝐴‘𝑠) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 )) |
18 | 17 | rexlimdva 3013 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 )) |
19 | | simpr 476 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) |
20 | | 0nn0 11184 |
. . . . . 6
⊢ 0 ∈
ℕ0 |
21 | | eqid 2610 |
. . . . . . 7
⊢
(coe1‘𝑀) = (coe1‘𝑀) |
22 | 21, 7, 6, 4 | coe1fvalcl 19403 |
. . . . . 6
⊢ ((𝑀 ∈ 𝐵 ∧ 0 ∈ ℕ0) →
((coe1‘𝑀)‘0) ∈ 𝐾) |
23 | 19, 20, 22 | sylancl 693 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ 𝐾) |
24 | 23 | adantr 480 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) →
((coe1‘𝑀)‘0) ∈ 𝐾) |
25 | | fveq2 6103 |
. . . . . 6
⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝐴‘𝑠) = (𝐴‘((coe1‘𝑀)‘0))) |
26 | 25 | eqeq2d 2620 |
. . . . 5
⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝑀 = (𝐴‘𝑠) ↔ 𝑀 = (𝐴‘((coe1‘𝑀)‘0)))) |
27 | 26 | adantl 481 |
. . . 4
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) ∧ 𝑠 = ((coe1‘𝑀)‘0)) → (𝑀 = (𝐴‘𝑠) ↔ 𝑀 = (𝐴‘((coe1‘𝑀)‘0)))) |
28 | | eqid 2610 |
. . . . . . . . . 10
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
29 | 6 | ply1ring 19439 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
30 | 6 | ply1lmod 19443 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
31 | | eqid 2610 |
. . . . . . . . . 10
⊢
(Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) |
32 | 8, 28, 29, 30, 31, 7 | asclf 19158 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵) |
33 | 32 | adantr 480 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵) |
34 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(Base‘𝑅) =
(Base‘𝑅) |
35 | 21, 7, 6, 34 | coe1fvalcl 19403 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ 𝐵 ∧ 0 ∈ ℕ0) →
((coe1‘𝑀)‘0) ∈ (Base‘𝑅)) |
36 | 19, 20, 35 | sylancl 693 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈
(Base‘𝑅)) |
37 | 6 | ply1sca 19444 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
38 | 37 | eqcomd 2616 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ Ring →
(Scalar‘𝑃) = 𝑅) |
39 | 38 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Ring →
(Base‘(Scalar‘𝑃)) = (Base‘𝑅)) |
40 | 39 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅)) |
41 | 36, 40 | eleqtrrd 2691 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈
(Base‘(Scalar‘𝑃))) |
42 | 33, 41 | ffvelrnd 6268 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵) |
43 | 1, 19, 42 | 3jca 1235 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵)) |
44 | 43 | adantr 480 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵)) |
45 | | simpr 476 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧
((coe1‘𝑀)‘𝑛) = 0 ) →
((coe1‘𝑀)‘𝑛) = 0 ) |
46 | 6, 8, 4, 5 | coe1scl 19478 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧
((coe1‘𝑀)‘0) ∈ 𝐾) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0
↦ if(𝑘 = 0,
((coe1‘𝑀)‘0), 0 ))) |
47 | 23, 46 | syldan 486 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0
↦ if(𝑘 = 0,
((coe1‘𝑀)‘0), 0 ))) |
48 | 47 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) →
(coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0
↦ if(𝑘 = 0,
((coe1‘𝑀)‘0), 0 ))) |
49 | | nnne0 10930 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) |
50 | 49 | neneqd 2787 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → ¬
𝑛 = 0) |
51 | 50 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0) |
52 | 51 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑛 = 0) |
53 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0)) |
54 | 53 | notbid 307 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0)) |
55 | 54 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0)) |
56 | 52, 55 | mpbird 246 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑘 = 0) |
57 | 56 | iffalsed 4047 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 ) = 0 ) |
58 | | nnnn0 11176 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
59 | 58 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
60 | | fvex 6113 |
. . . . . . . . . . . . . . 15
⊢
(0g‘𝑅) ∈ V |
61 | 5, 60 | eqeltri 2684 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
62 | 61 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 ∈ V) |
63 | 48, 57, 59, 62 | fvmptd 6197 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) →
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = 0 ) |
64 | 63 | eqcomd 2616 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
65 | 64 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧
((coe1‘𝑀)‘𝑛) = 0 ) → 0 =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
66 | 45, 65 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧
((coe1‘𝑀)‘𝑛) = 0 ) →
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
67 | 66 | ex 449 |
. . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) →
(((coe1‘𝑀)‘𝑛) = 0 →
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))) |
68 | 67 | ralimdva 2945 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 → ∀𝑛 ∈ ℕ
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))) |
69 | 68 | imp 444 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
70 | 6, 8, 4 | ply1sclid 19479 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧
((coe1‘𝑀)‘0) ∈ 𝐾) → ((coe1‘𝑀)‘0) =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)) |
71 | 23, 70 | syldan 486 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)) |
72 | 71 | adantr 480 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) →
((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)) |
73 | | df-n0 11170 |
. . . . . . . 8
⊢
ℕ0 = (ℕ ∪ {0}) |
74 | 73 | raleqi 3119 |
. . . . . . 7
⊢
(∀𝑛 ∈
ℕ0 ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ ∀𝑛 ∈ (ℕ ∪
{0})((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
75 | | c0ex 9913 |
. . . . . . . 8
⊢ 0 ∈
V |
76 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑛 = 0 →
((coe1‘𝑀)‘𝑛) = ((coe1‘𝑀)‘0)) |
77 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑛 = 0 →
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) =
((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)) |
78 | 76, 77 | eqeq12d 2625 |
. . . . . . . . 9
⊢ (𝑛 = 0 →
(((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔
((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))) |
79 | 78 | ralunsn 4360 |
. . . . . . . 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)))) |
80 | 75, 79 | 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)))) |
81 | 74, 80 | syl5bb 271 |
. . . . . 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)))) |
82 | 69, 72, 81 | mpbir2and 959 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ0
((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)) |
83 | | eqid 2610 |
. . . . . 6
⊢
(coe1‘(𝐴‘((coe1‘𝑀)‘0))) =
(coe1‘(𝐴‘((coe1‘𝑀)‘0))) |
84 | 6, 7, 21, 83 | eqcoe1ply1eq 19488 |
. . . . 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)))) |
85 | 44, 82, 84 | sylc 63 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → 𝑀 = (𝐴‘((coe1‘𝑀)‘0))) |
86 | 24, 27, 85 | rspcedvd 3289 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠)) |
87 | 86 | ex 449 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠))) |
88 | 18, 87 | impbid 201 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 )) |