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 )