Metamath Proof Explorer


Theorem chpmat0d

Description: The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019)

Ref Expression
Hypothesis chpmat0.c 𝐶 = ( ∅ CharPlyMat 𝑅 )
Assertion chpmat0d ( 𝑅 ∈ Ring → ( 𝐶 ‘ ∅ ) = ( 1r ‘ ( Poly1𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 chpmat0.c 𝐶 = ( ∅ CharPlyMat 𝑅 )
2 0fin ∅ ∈ Fin
3 id ( 𝑅 ∈ Ring → 𝑅 ∈ Ring )
4 0ex ∅ ∈ V
5 4 snid ∅ ∈ { ∅ }
6 mat0dimbas0 ( 𝑅 ∈ Ring → ( Base ‘ ( ∅ Mat 𝑅 ) ) = { ∅ } )
7 5 6 eleqtrrid ( 𝑅 ∈ Ring → ∅ ∈ ( Base ‘ ( ∅ Mat 𝑅 ) ) )
8 eqid ( ∅ Mat 𝑅 ) = ( ∅ Mat 𝑅 )
9 eqid ( Base ‘ ( ∅ Mat 𝑅 ) ) = ( Base ‘ ( ∅ Mat 𝑅 ) )
10 eqid ( Poly1𝑅 ) = ( Poly1𝑅 )
11 eqid ( ∅ Mat ( Poly1𝑅 ) ) = ( ∅ Mat ( Poly1𝑅 ) )
12 eqid ( ∅ maDet ( Poly1𝑅 ) ) = ( ∅ maDet ( Poly1𝑅 ) )
13 eqid ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) )
14 eqid ( var1𝑅 ) = ( var1𝑅 )
15 eqid ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) )
16 eqid ( ∅ matToPolyMat 𝑅 ) = ( ∅ matToPolyMat 𝑅 )
17 eqid ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) )
18 1 8 9 10 11 12 13 14 15 16 17 chpmatval ( ( ∅ ∈ Fin ∧ 𝑅 ∈ Ring ∧ ∅ ∈ ( Base ‘ ( ∅ Mat 𝑅 ) ) ) → ( 𝐶 ‘ ∅ ) = ( ( ∅ maDet ( Poly1𝑅 ) ) ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) )
19 2 3 7 18 mp3an2i ( 𝑅 ∈ Ring → ( 𝐶 ‘ ∅ ) = ( ( ∅ maDet ( Poly1𝑅 ) ) ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) )
20 10 ply1ring ( 𝑅 ∈ Ring → ( Poly1𝑅 ) ∈ Ring )
21 mdet0pr ( ( Poly1𝑅 ) ∈ Ring → ( ∅ maDet ( Poly1𝑅 ) ) = { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } )
22 21 fveq1d ( ( Poly1𝑅 ) ∈ Ring → ( ( ∅ maDet ( Poly1𝑅 ) ) ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) = ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) )
23 20 22 syl ( 𝑅 ∈ Ring → ( ( ∅ maDet ( Poly1𝑅 ) ) ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) = ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) )
24 11 mat0dimid ( ( Poly1𝑅 ) ∈ Ring → ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ∅ )
25 20 24 syl ( 𝑅 ∈ Ring → ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ∅ )
26 25 oveq2d ( 𝑅 ∈ Ring → ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) = ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ∅ ) )
27 eqid ( Base ‘ ( Poly1𝑅 ) ) = ( Base ‘ ( Poly1𝑅 ) )
28 14 10 27 vr1cl ( 𝑅 ∈ Ring → ( var1𝑅 ) ∈ ( Base ‘ ( Poly1𝑅 ) ) )
29 11 mat0dimscm ( ( ( Poly1𝑅 ) ∈ Ring ∧ ( var1𝑅 ) ∈ ( Base ‘ ( Poly1𝑅 ) ) ) → ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ∅ ) = ∅ )
30 20 28 29 syl2anc ( 𝑅 ∈ Ring → ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ∅ ) = ∅ )
31 26 30 eqtrd ( 𝑅 ∈ Ring → ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) = ∅ )
32 d0mat2pmat ( 𝑅 ∈ Ring → ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) = ∅ )
33 31 32 oveq12d ( 𝑅 ∈ Ring → ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) = ( ∅ ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ∅ ) )
34 11 matring ( ( ∅ ∈ Fin ∧ ( Poly1𝑅 ) ∈ Ring ) → ( ∅ Mat ( Poly1𝑅 ) ) ∈ Ring )
35 2 20 34 sylancr ( 𝑅 ∈ Ring → ( ∅ Mat ( Poly1𝑅 ) ) ∈ Ring )
36 ringgrp ( ( ∅ Mat ( Poly1𝑅 ) ) ∈ Ring → ( ∅ Mat ( Poly1𝑅 ) ) ∈ Grp )
37 35 36 syl ( 𝑅 ∈ Ring → ( ∅ Mat ( Poly1𝑅 ) ) ∈ Grp )
38 mat0dimbas0 ( ( Poly1𝑅 ) ∈ Ring → ( Base ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = { ∅ } )
39 20 38 syl ( 𝑅 ∈ Ring → ( Base ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = { ∅ } )
40 5 39 eleqtrrid ( 𝑅 ∈ Ring → ∅ ∈ ( Base ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) )
41 eqid ( Base ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ( Base ‘ ( ∅ Mat ( Poly1𝑅 ) ) )
42 eqid ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) )
43 41 42 13 grpsubid ( ( ( ∅ Mat ( Poly1𝑅 ) ) ∈ Grp ∧ ∅ ∈ ( Base ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) → ( ∅ ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ∅ ) = ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) )
44 37 40 43 syl2anc ( 𝑅 ∈ Ring → ( ∅ ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ∅ ) = ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) )
45 33 44 eqtrd ( 𝑅 ∈ Ring → ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) = ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) )
46 45 fveq2d ( 𝑅 ∈ Ring → ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) = ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) )
47 11 mat0dim0 ( ( Poly1𝑅 ) ∈ Ring → ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ∅ )
48 20 47 syl ( 𝑅 ∈ Ring → ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) = ∅ )
49 48 fveq2d ( 𝑅 ∈ Ring → ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) = ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ∅ ) )
50 fvex ( 1r ‘ ( Poly1𝑅 ) ) ∈ V
51 4 50 fvsn ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ∅ ) = ( 1r ‘ ( Poly1𝑅 ) )
52 49 51 eqtrdi ( 𝑅 ∈ Ring → ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ( 0g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) = ( 1r ‘ ( Poly1𝑅 ) ) )
53 46 52 eqtrd ( 𝑅 ∈ Ring → ( { ⟨ ∅ , ( 1r ‘ ( Poly1𝑅 ) ) ⟩ } ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) = ( 1r ‘ ( Poly1𝑅 ) ) )
54 23 53 eqtrd ( 𝑅 ∈ Ring → ( ( ∅ maDet ( Poly1𝑅 ) ) ‘ ( ( ( var1𝑅 ) ( ·𝑠 ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( 1r ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ) ( -g ‘ ( ∅ Mat ( Poly1𝑅 ) ) ) ( ( ∅ matToPolyMat 𝑅 ) ‘ ∅ ) ) ) = ( 1r ‘ ( Poly1𝑅 ) ) )
55 19 54 eqtrd ( 𝑅 ∈ Ring → ( 𝐶 ‘ ∅ ) = ( 1r ‘ ( Poly1𝑅 ) ) )