Metamath Proof Explorer

Theorem chpmatply1

Description: The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in Lang, p. 561: "[the characteristic polynomial] is an element of k[t". (Contributed by AV, 2-Aug-2019) (Proof shortened by AV, 29-Nov-2019)

Ref Expression
Hypotheses chpmatply1.c ⊢ 𝐶 = ( 𝑁 CharPlyMat 𝑅 )
chpmatply1.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
chpmatply1.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
chpmatply1.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
chpmatply1.e ⊢ 𝐸 = ( Base ‘ 𝑃 )
Assertion chpmatply1 ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝐶 ‘ 𝑀 ) ∈ 𝐸 )

Proof

Step Hyp Ref Expression
1 chpmatply1.c ⊢ 𝐶 = ( 𝑁 CharPlyMat 𝑅 )
2 chpmatply1.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
3 chpmatply1.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
4 chpmatply1.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
5 chpmatply1.e ⊢ 𝐸 = ( Base ‘ 𝑃 )
6 eqid ⊢ ( 𝑁 Mat 𝑃 ) = ( 𝑁 Mat 𝑃 )
7 eqid ⊢ ( 𝑁 maDet 𝑃 ) = ( 𝑁 maDet 𝑃 )
8 eqid ⊢ ( -g ‘ ( 𝑁 Mat 𝑃 ) ) = ( -g ‘ ( 𝑁 Mat 𝑃 ) )
9 eqid ⊢ ( var1 ‘ 𝑅 ) = ( var1 ‘ 𝑅 )
10 eqid ⊢ ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) = ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) )
11 eqid ⊢ ( 𝑁 matToPolyMat 𝑅 ) = ( 𝑁 matToPolyMat 𝑅 )
12 eqid ⊢ ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) = ( 1r ‘ ( 𝑁 Mat 𝑃 ) )
13 1 2 3 4 6 7 8 9 10 11 12 chpmatval ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝐶 ‘ 𝑀 ) = ( ( 𝑁 maDet 𝑃 ) ‘ ( ( ( var1 ‘ 𝑅 ) ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) ) ( -g ‘ ( 𝑁 Mat 𝑃 ) ) ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 𝑀 ) ) ) )
14 4 ply1crng ⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ CRing )
15 14 3ad2ant2 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑃 ∈ CRing )
16 crngring ⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring )
17 eqid ⊢ ( ( ( var1 ‘ 𝑅 ) ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) ) ( -g ‘ ( 𝑁 Mat 𝑃 ) ) ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 𝑀 ) ) = ( ( ( var1 ‘ 𝑅 ) ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) ) ( -g ‘ ( 𝑁 Mat 𝑃 ) ) ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 𝑀 ) )
18 2 3 4 6 9 11 8 10 12 17 chmatcl ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( ( ( var1 ‘ 𝑅 ) ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) ) ( -g ‘ ( 𝑁 Mat 𝑃 ) ) ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 𝑀 ) ) ∈ ( Base ‘ ( 𝑁 Mat 𝑃 ) ) )
19 16 18 syl3an2 ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( ( ( var1 ‘ 𝑅 ) ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) ) ( -g ‘ ( 𝑁 Mat 𝑃 ) ) ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 𝑀 ) ) ∈ ( Base ‘ ( 𝑁 Mat 𝑃 ) ) )
20 eqid ⊢ ( Base ‘ ( 𝑁 Mat 𝑃 ) ) = ( Base ‘ ( 𝑁 Mat 𝑃 ) )
21 7 6 20 5 mdetcl ⊢ ( ( 𝑃 ∈ CRing ∧ ( ( ( var1 ‘ 𝑅 ) ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) ) ( -g ‘ ( 𝑁 Mat 𝑃 ) ) ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 𝑀 ) ) ∈ ( Base ‘ ( 𝑁 Mat 𝑃 ) ) ) → ( ( 𝑁 maDet 𝑃 ) ‘ ( ( ( var1 ‘ 𝑅 ) ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) ) ( -g ‘ ( 𝑁 Mat 𝑃 ) ) ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 𝑀 ) ) ) ∈ 𝐸 )
22 15 19 21 syl2anc ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( ( 𝑁 maDet 𝑃 ) ‘ ( ( ( var1 ‘ 𝑅 ) ( ·𝑠 ‘ ( 𝑁 Mat 𝑃 ) ) ( 1r ‘ ( 𝑁 Mat 𝑃 ) ) ) ( -g ‘ ( 𝑁 Mat 𝑃 ) ) ( ( 𝑁 matToPolyMat 𝑅 ) ‘ 𝑀 ) ) ) ∈ 𝐸 )
23 13 22 eqeltrd ⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝐶 ‘ 𝑀 ) ∈ 𝐸 )