Step | Hyp | Ref
| Expression |
1 | | cply1coe0.k |
. . . . . 6
β’ πΎ = (Baseβπ
) |
2 | | cply1coe0.0 |
. . . . . 6
β’ 0 =
(0gβπ
) |
3 | | cply1coe0.p |
. . . . . 6
β’ π = (Poly1βπ
) |
4 | | cply1coe0.b |
. . . . . 6
β’ π΅ = (Baseβπ) |
5 | | cply1coe0.a |
. . . . . 6
β’ π΄ = (algScβπ) |
6 | 1, 2, 3, 4, 5 | cply1coe0 21686 |
. . . . 5
β’ ((π
β Ring β§ π β πΎ) β βπ β β
((coe1β(π΄βπ ))βπ) = 0 ) |
7 | 6 | ad4ant13 750 |
. . . 4
β’ ((((π
β Ring β§ π β π΅) β§ π β πΎ) β§ π = (π΄βπ )) β βπ β β
((coe1β(π΄βπ ))βπ) = 0 ) |
8 | | fveq2 6843 |
. . . . . . . 8
β’ (π = (π΄βπ ) β (coe1βπ) =
(coe1β(π΄βπ ))) |
9 | 8 | fveq1d 6845 |
. . . . . . 7
β’ (π = (π΄βπ ) β ((coe1βπ)βπ) = ((coe1β(π΄βπ ))βπ)) |
10 | 9 | eqeq1d 2735 |
. . . . . 6
β’ (π = (π΄βπ ) β (((coe1βπ)βπ) = 0 β
((coe1β(π΄βπ ))βπ) = 0 )) |
11 | 10 | ralbidv 3171 |
. . . . 5
β’ (π = (π΄βπ ) β (βπ β β ((coe1βπ)βπ) = 0 β βπ β β
((coe1β(π΄βπ ))βπ) = 0 )) |
12 | 11 | adantl 483 |
. . . 4
β’ ((((π
β Ring β§ π β π΅) β§ π β πΎ) β§ π = (π΄βπ )) β (βπ β β ((coe1βπ)βπ) = 0 β βπ β β
((coe1β(π΄βπ ))βπ) = 0 )) |
13 | 7, 12 | mpbird 257 |
. . 3
β’ ((((π
β Ring β§ π β π΅) β§ π β πΎ) β§ π = (π΄βπ )) β βπ β β ((coe1βπ)βπ) = 0 ) |
14 | 13 | rexlimdva2 3151 |
. 2
β’ ((π
β Ring β§ π β π΅) β (βπ β πΎ π = (π΄βπ ) β βπ β β ((coe1βπ)βπ) = 0 )) |
15 | | simpr 486 |
. . . . . 6
β’ ((π
β Ring β§ π β π΅) β π β π΅) |
16 | | 0nn0 12433 |
. . . . . 6
β’ 0 β
β0 |
17 | | eqid 2733 |
. . . . . . 7
β’
(coe1βπ) = (coe1βπ) |
18 | 17, 4, 3, 1 | coe1fvalcl 21599 |
. . . . . 6
β’ ((π β π΅ β§ 0 β β0) β
((coe1βπ)β0) β πΎ) |
19 | 15, 16, 18 | sylancl 587 |
. . . . 5
β’ ((π
β Ring β§ π β π΅) β ((coe1βπ)β0) β πΎ) |
20 | 19 | adantr 482 |
. . . 4
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β
((coe1βπ)β0) β πΎ) |
21 | | fveq2 6843 |
. . . . . 6
β’ (π = ((coe1βπ)β0) β (π΄βπ ) = (π΄β((coe1βπ)β0))) |
22 | 21 | eqeq2d 2744 |
. . . . 5
β’ (π = ((coe1βπ)β0) β (π = (π΄βπ ) β π = (π΄β((coe1βπ)β0)))) |
23 | 22 | adantl 483 |
. . . 4
β’ ((((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β§ π = ((coe1βπ)β0)) β (π = (π΄βπ ) β π = (π΄β((coe1βπ)β0)))) |
24 | | simpl 484 |
. . . . . . 7
β’ ((π
β Ring β§ π β π΅) β π
β Ring) |
25 | | eqid 2733 |
. . . . . . . . . 10
β’
(Scalarβπ) =
(Scalarβπ) |
26 | 3 | ply1ring 21635 |
. . . . . . . . . 10
β’ (π
β Ring β π β Ring) |
27 | 3 | ply1lmod 21639 |
. . . . . . . . . 10
β’ (π
β Ring β π β LMod) |
28 | | eqid 2733 |
. . . . . . . . . 10
β’
(Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) |
29 | 5, 25, 26, 27, 28, 4 | asclf 21301 |
. . . . . . . . 9
β’ (π
β Ring β π΄:(Baseβ(Scalarβπ))βΆπ΅) |
30 | 29 | adantr 482 |
. . . . . . . 8
β’ ((π
β Ring β§ π β π΅) β π΄:(Baseβ(Scalarβπ))βΆπ΅) |
31 | | eqid 2733 |
. . . . . . . . . . 11
β’
(Baseβπ
) =
(Baseβπ
) |
32 | 17, 4, 3, 31 | coe1fvalcl 21599 |
. . . . . . . . . 10
β’ ((π β π΅ β§ 0 β β0) β
((coe1βπ)β0) β (Baseβπ
)) |
33 | 15, 16, 32 | sylancl 587 |
. . . . . . . . 9
β’ ((π
β Ring β§ π β π΅) β ((coe1βπ)β0) β
(Baseβπ
)) |
34 | 3 | ply1sca 21640 |
. . . . . . . . . . . 12
β’ (π
β Ring β π
= (Scalarβπ)) |
35 | 34 | eqcomd 2739 |
. . . . . . . . . . 11
β’ (π
β Ring β
(Scalarβπ) = π
) |
36 | 35 | fveq2d 6847 |
. . . . . . . . . 10
β’ (π
β Ring β
(Baseβ(Scalarβπ)) = (Baseβπ
)) |
37 | 36 | adantr 482 |
. . . . . . . . 9
β’ ((π
β Ring β§ π β π΅) β (Baseβ(Scalarβπ)) = (Baseβπ
)) |
38 | 33, 37 | eleqtrrd 2837 |
. . . . . . . 8
β’ ((π
β Ring β§ π β π΅) β ((coe1βπ)β0) β
(Baseβ(Scalarβπ))) |
39 | 30, 38 | ffvelcdmd 7037 |
. . . . . . 7
β’ ((π
β Ring β§ π β π΅) β (π΄β((coe1βπ)β0)) β π΅) |
40 | 24, 15, 39 | 3jca 1129 |
. . . . . 6
β’ ((π
β Ring β§ π β π΅) β (π
β Ring β§ π β π΅ β§ (π΄β((coe1βπ)β0)) β π΅)) |
41 | 40 | adantr 482 |
. . . . 5
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β (π
β Ring β§ π β π΅ β§ (π΄β((coe1βπ)β0)) β π΅)) |
42 | | simpr 486 |
. . . . . . . . . 10
β’ ((((π
β Ring β§ π β π΅) β§ π β β) β§
((coe1βπ)βπ) = 0 ) β
((coe1βπ)βπ) = 0 ) |
43 | 3, 5, 1, 2 | coe1scl 21674 |
. . . . . . . . . . . . . . 15
β’ ((π
β Ring β§
((coe1βπ)β0) β πΎ) β (coe1β(π΄β((coe1βπ)β0))) = (π β β0
β¦ if(π = 0,
((coe1βπ)β0), 0 ))) |
44 | 19, 43 | syldan 592 |
. . . . . . . . . . . . . 14
β’ ((π
β Ring β§ π β π΅) β (coe1β(π΄β((coe1βπ)β0))) = (π β β0
β¦ if(π = 0,
((coe1βπ)β0), 0 ))) |
45 | 44 | adantr 482 |
. . . . . . . . . . . . 13
β’ (((π
β Ring β§ π β π΅) β§ π β β) β
(coe1β(π΄β((coe1βπ)β0))) = (π β β0
β¦ if(π = 0,
((coe1βπ)β0), 0 ))) |
46 | | nnne0 12192 |
. . . . . . . . . . . . . . . . . 18
β’ (π β β β π β 0) |
47 | 46 | neneqd 2945 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β Β¬
π = 0) |
48 | 47 | adantl 483 |
. . . . . . . . . . . . . . . 16
β’ (((π
β Ring β§ π β π΅) β§ π β β) β Β¬ π = 0) |
49 | 48 | adantr 482 |
. . . . . . . . . . . . . . 15
β’ ((((π
β Ring β§ π β π΅) β§ π β β) β§ π = π) β Β¬ π = 0) |
50 | | eqeq1 2737 |
. . . . . . . . . . . . . . . . 17
β’ (π = π β (π = 0 β π = 0)) |
51 | 50 | notbid 318 |
. . . . . . . . . . . . . . . 16
β’ (π = π β (Β¬ π = 0 β Β¬ π = 0)) |
52 | 51 | adantl 483 |
. . . . . . . . . . . . . . 15
β’ ((((π
β Ring β§ π β π΅) β§ π β β) β§ π = π) β (Β¬ π = 0 β Β¬ π = 0)) |
53 | 49, 52 | mpbird 257 |
. . . . . . . . . . . . . 14
β’ ((((π
β Ring β§ π β π΅) β§ π β β) β§ π = π) β Β¬ π = 0) |
54 | 53 | iffalsed 4498 |
. . . . . . . . . . . . 13
β’ ((((π
β Ring β§ π β π΅) β§ π β β) β§ π = π) β if(π = 0, ((coe1βπ)β0), 0 ) = 0 ) |
55 | | nnnn0 12425 |
. . . . . . . . . . . . . 14
β’ (π β β β π β
β0) |
56 | 55 | adantl 483 |
. . . . . . . . . . . . 13
β’ (((π
β Ring β§ π β π΅) β§ π β β) β π β β0) |
57 | 2 | fvexi 6857 |
. . . . . . . . . . . . . 14
β’ 0 β
V |
58 | 57 | a1i 11 |
. . . . . . . . . . . . 13
β’ (((π
β Ring β§ π β π΅) β§ π β β) β 0 β V) |
59 | 45, 54, 56, 58 | fvmptd 6956 |
. . . . . . . . . . . 12
β’ (((π
β Ring β§ π β π΅) β§ π β β) β
((coe1β(π΄β((coe1βπ)β0)))βπ) = 0 ) |
60 | 59 | eqcomd 2739 |
. . . . . . . . . . 11
β’ (((π
β Ring β§ π β π΅) β§ π β β) β 0 =
((coe1β(π΄β((coe1βπ)β0)))βπ)) |
61 | 60 | adantr 482 |
. . . . . . . . . 10
β’ ((((π
β Ring β§ π β π΅) β§ π β β) β§
((coe1βπ)βπ) = 0 ) β 0 =
((coe1β(π΄β((coe1βπ)β0)))βπ)) |
62 | 42, 61 | eqtrd 2773 |
. . . . . . . . 9
β’ ((((π
β Ring β§ π β π΅) β§ π β β) β§
((coe1βπ)βπ) = 0 ) β
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ)) |
63 | 62 | ex 414 |
. . . . . . . 8
β’ (((π
β Ring β§ π β π΅) β§ π β β) β
(((coe1βπ)βπ) = 0 β
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ))) |
64 | 63 | ralimdva 3161 |
. . . . . . 7
β’ ((π
β Ring β§ π β π΅) β (βπ β β ((coe1βπ)βπ) = 0 β βπ β β
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ))) |
65 | 64 | imp 408 |
. . . . . 6
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β βπ β β
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ)) |
66 | 3, 5, 1 | ply1sclid 21675 |
. . . . . . . 8
β’ ((π
β Ring β§
((coe1βπ)β0) β πΎ) β ((coe1βπ)β0) =
((coe1β(π΄β((coe1βπ)β0)))β0)) |
67 | 19, 66 | syldan 592 |
. . . . . . 7
β’ ((π
β Ring β§ π β π΅) β ((coe1βπ)β0) =
((coe1β(π΄β((coe1βπ)β0)))β0)) |
68 | 67 | adantr 482 |
. . . . . 6
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β
((coe1βπ)β0) = ((coe1β(π΄β((coe1βπ)β0)))β0)) |
69 | | df-n0 12419 |
. . . . . . . 8
β’
β0 = (β βͺ {0}) |
70 | 69 | raleqi 3310 |
. . . . . . 7
β’
(βπ β
β0 ((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β βπ β (β βͺ
{0})((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ)) |
71 | | c0ex 11154 |
. . . . . . . 8
β’ 0 β
V |
72 | | fveq2 6843 |
. . . . . . . . . 10
β’ (π = 0 β
((coe1βπ)βπ) = ((coe1βπ)β0)) |
73 | | fveq2 6843 |
. . . . . . . . . 10
β’ (π = 0 β
((coe1β(π΄β((coe1βπ)β0)))βπ) =
((coe1β(π΄β((coe1βπ)β0)))β0)) |
74 | 72, 73 | eqeq12d 2749 |
. . . . . . . . 9
β’ (π = 0 β
(((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β
((coe1βπ)β0) = ((coe1β(π΄β((coe1βπ)β0)))β0))) |
75 | 74 | ralunsn 4852 |
. . . . . . . 8
β’ (0 β
V β (βπ β
(β βͺ {0})((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β (βπ β β
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β§
((coe1βπ)β0) = ((coe1β(π΄β((coe1βπ)β0)))β0)))) |
76 | 71, 75 | mp1i 13 |
. . . . . . 7
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β (βπ β (β βͺ
{0})((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β (βπ β β
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β§
((coe1βπ)β0) = ((coe1β(π΄β((coe1βπ)β0)))β0)))) |
77 | 70, 76 | bitrid 283 |
. . . . . 6
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β (βπ β β0
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β (βπ β β
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β§
((coe1βπ)β0) = ((coe1β(π΄β((coe1βπ)β0)))β0)))) |
78 | 65, 68, 77 | mpbir2and 712 |
. . . . 5
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β βπ β β0
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ)) |
79 | | eqid 2733 |
. . . . . 6
β’
(coe1β(π΄β((coe1βπ)β0))) =
(coe1β(π΄β((coe1βπ)β0))) |
80 | 3, 4, 17, 79 | eqcoe1ply1eq 21684 |
. . . . 5
β’ ((π
β Ring β§ π β π΅ β§ (π΄β((coe1βπ)β0)) β π΅) β (βπ β β0
((coe1βπ)βπ) = ((coe1β(π΄β((coe1βπ)β0)))βπ) β π = (π΄β((coe1βπ)β0)))) |
81 | 41, 78, 80 | sylc 65 |
. . . 4
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β π = (π΄β((coe1βπ)β0))) |
82 | 20, 23, 81 | rspcedvd 3582 |
. . 3
β’ (((π
β Ring β§ π β π΅) β§ βπ β β ((coe1βπ)βπ) = 0 ) β βπ β πΎ π = (π΄βπ )) |
83 | 82 | ex 414 |
. 2
β’ ((π
β Ring β§ π β π΅) β (βπ β β ((coe1βπ)βπ) = 0 β βπ β πΎ π = (π΄βπ ))) |
84 | 14, 83 | impbid 211 |
1
β’ ((π
β Ring β§ π β π΅) β (βπ β πΎ π = (π΄βπ ) β βπ β β ((coe1βπ)βπ) = 0 )) |