Step | Hyp | Ref
| Expression |
1 | | chpdmat.c |
. . . 4
β’ πΆ = (π CharPlyMat π
) |
2 | | chpdmat.a |
. . . 4
β’ π΄ = (π Mat π
) |
3 | | chpdmat.b |
. . . 4
β’ π΅ = (Baseβπ΄) |
4 | | chpdmat.p |
. . . 4
β’ π = (Poly1βπ
) |
5 | | eqid 2733 |
. . . 4
β’ (π Mat π) = (π Mat π) |
6 | | eqid 2733 |
. . . 4
β’ (π maDet π) = (π maDet π) |
7 | | eqid 2733 |
. . . 4
β’
(-gβ(π Mat π)) = (-gβ(π Mat π)) |
8 | | chpdmat.x |
. . . 4
β’ π = (var1βπ
) |
9 | | eqid 2733 |
. . . 4
β’ (
Β·π β(π Mat π)) = ( Β·π
β(π Mat π)) |
10 | | eqid 2733 |
. . . 4
β’ (π matToPolyMat π
) = (π matToPolyMat π
) |
11 | | eqid 2733 |
. . . 4
β’
(1rβ(π Mat π)) = (1rβ(π Mat π)) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 | chpmatval 22203 |
. . 3
β’ ((π β Fin β§ π
β CRing β§ π β π΅) β (πΆβπ) = ((π maDet π)β((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ)))) |
13 | 12 | adantr 482 |
. 2
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ βπ β π βπ β π (π β π β (πππ) = 0 )) β (πΆβπ) = ((π maDet π)β((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ)))) |
14 | 4 | ply1crng 21592 |
. . . . . 6
β’ (π
β CRing β π β CRing) |
15 | 14 | 3ad2ant2 1135 |
. . . . 5
β’ ((π β Fin β§ π
β CRing β§ π β π΅) β π β CRing) |
16 | | simp1 1137 |
. . . . 5
β’ ((π β Fin β§ π
β CRing β§ π β π΅) β π β Fin) |
17 | | crngring 19984 |
. . . . . . 7
β’ (π
β CRing β π
β Ring) |
18 | 17 | 3anim2i 1154 |
. . . . . 6
β’ ((π β Fin β§ π
β CRing β§ π β π΅) β (π β Fin β§ π
β Ring β§ π β π΅)) |
19 | | chpdmat.s |
. . . . . . 7
β’ π = (algScβπ) |
20 | | chpdmat.0 |
. . . . . . 7
β’ 0 =
(0gβπ
) |
21 | | chpdmat.g |
. . . . . . 7
β’ πΊ = (mulGrpβπ) |
22 | | chpdmat.m |
. . . . . . 7
β’ β =
(-gβπ) |
23 | 1, 4, 2, 19, 3, 8,
20, 21, 22, 5, 11, 9, 7,
10 | chpdmatlem1 22210 |
. . . . . 6
β’ ((π β Fin β§ π
β Ring β§ π β π΅) β ((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ)) β (Baseβ(π Mat π))) |
24 | 18, 23 | syl 17 |
. . . . 5
β’ ((π β Fin β§ π
β CRing β§ π β π΅) β ((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ)) β (Baseβ(π Mat π))) |
25 | 15, 16, 24 | 3jca 1129 |
. . . 4
β’ ((π β Fin β§ π
β CRing β§ π β π΅) β (π β CRing β§ π β Fin β§ ((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ)) β (Baseβ(π Mat π)))) |
26 | 25 | adantr 482 |
. . 3
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ βπ β π βπ β π (π β π β (πππ) = 0 )) β (π β CRing β§ π β Fin β§ ((π(
Β·π β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ)) β (Baseβ(π Mat π)))) |
27 | 18 | anim1i 616 |
. . . . . . . . . 10
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ π β π) β ((π β Fin β§ π
β Ring β§ π β π΅) β§ π β π)) |
28 | 27 | anim1i 616 |
. . . . . . . . 9
β’ ((((π β Fin β§ π
β CRing β§ π β π΅) β§ π β π) β§ π β π) β (((π β Fin β§ π
β Ring β§ π β π΅) β§ π β π) β§ π β π)) |
29 | 1, 4, 2, 19, 3, 8,
20, 21, 22, 5, 11, 9, 7,
10 | chpdmatlem2 22211 |
. . . . . . . . 9
β’
((((((π β Fin
β§ π
β Ring β§
π β π΅) β§ π β π) β§ π β π) β§ π β π) β§ (πππ) = 0 ) β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (0gβπ)) |
30 | 28, 29 | sylanl1 679 |
. . . . . . . 8
β’
((((((π β Fin
β§ π
β CRing β§
π β π΅) β§ π β π) β§ π β π) β§ π β π) β§ (πππ) = 0 ) β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (0gβπ)) |
31 | 30 | exp31 421 |
. . . . . . 7
β’ ((((π β Fin β§ π
β CRing β§ π β π΅) β§ π β π) β§ π β π) β (π β π β ((πππ) = 0 β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (0gβπ)))) |
32 | 31 | a2d 29 |
. . . . . 6
β’ ((((π β Fin β§ π
β CRing β§ π β π΅) β§ π β π) β§ π β π) β ((π β π β (πππ) = 0 ) β (π β π β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (0gβπ)))) |
33 | 32 | ralimdva 3161 |
. . . . 5
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ π β π) β (βπ β π (π β π β (πππ) = 0 ) β βπ β π (π β π β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (0gβπ)))) |
34 | 33 | ralimdva 3161 |
. . . 4
β’ ((π β Fin β§ π
β CRing β§ π β π΅) β (βπ β π βπ β π (π β π β (πππ) = 0 ) β βπ β π βπ β π (π β π β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (0gβπ)))) |
35 | 34 | imp 408 |
. . 3
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ βπ β π βπ β π (π β π β (πππ) = 0 )) β βπ β π βπ β π (π β π β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (0gβπ))) |
36 | | eqid 2733 |
. . . 4
β’
(Baseβ(π Mat
π)) = (Baseβ(π Mat π)) |
37 | | eqid 2733 |
. . . 4
β’
(0gβπ) = (0gβπ) |
38 | 6, 5, 36, 21, 37 | mdetdiag 21971 |
. . 3
β’ ((π β CRing β§ π β Fin β§ ((π(
Β·π β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ)) β (Baseβ(π Mat π))) β (βπ β π βπ β π (π β π β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (0gβπ)) β ((π maDet π)β((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))) = (πΊ Ξ£g (π β π β¦ (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π))))) |
39 | 26, 35, 38 | sylc 65 |
. 2
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ βπ β π βπ β π (π β π β (πππ) = 0 )) β ((π maDet π)β((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))) = (πΊ Ξ£g (π β π β¦ (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π)))) |
40 | 1, 4, 2, 19, 3, 8,
20, 21, 22, 5, 11, 9, 7,
10 | chpdmatlem3 22212 |
. . . . . 6
β’ (((π β Fin β§ π
β Ring β§ π β π΅) β§ π β π) β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (π β (πβ(πππ)))) |
41 | 18, 40 | sylan 581 |
. . . . 5
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ π β π) β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (π β (πβ(πππ)))) |
42 | 41 | adantlr 714 |
. . . 4
β’ ((((π β Fin β§ π
β CRing β§ π β π΅) β§ βπ β π βπ β π (π β π β (πππ) = 0 )) β§ π β π) β (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π) = (π β (πβ(πππ)))) |
43 | 42 | mpteq2dva 5209 |
. . 3
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ βπ β π βπ β π (π β π β (πππ) = 0 )) β (π β π β¦ (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π)) = (π β π β¦ (π β (πβ(πππ))))) |
44 | 43 | oveq2d 7377 |
. 2
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ βπ β π βπ β π (π β π β (πππ) = 0 )) β (πΊ Ξ£g
(π β π β¦ (π((π( Β·π
β(π Mat π))(1rβ(π Mat π)))(-gβ(π Mat π))((π matToPolyMat π
)βπ))π))) = (πΊ Ξ£g (π β π β¦ (π β (πβ(πππ)))))) |
45 | 13, 39, 44 | 3eqtrd 2777 |
1
β’ (((π β Fin β§ π
β CRing β§ π β π΅) β§ βπ β π βπ β π (π β π β (πππ) = 0 )) β (πΆβπ) = (πΊ Ξ£g (π β π β¦ (π β (πβ(πππ)))))) |