Theorem cpmidg2sum

Description: Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019)

	Ref	Expression
Hypotheses	cpmadugsum.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	cpmadugsum.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	cpmadugsum.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	cpmadugsum.y	⊢ 𝑌 = ( 𝑁 Mat 𝑃 )
	cpmadugsum.t	⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 )
	cpmadugsum.x	⊢ 𝑋 = ( var1 ‘ 𝑅 )
	cpmadugsum.e	⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑃 ) )
	cpmadugsum.m	⊢ · = ( ·𝑠 ‘ 𝑌 )
	cpmadugsum.r	⊢ × = ( .r ‘ 𝑌 )
	cpmadugsum.1	⊢ 1 = ( 1r ‘ 𝑌 )
	cpmadugsum.g	⊢ + = ( +g ‘ 𝑌 )
	cpmadugsum.s	⊢ − = ( -g ‘ 𝑌 )
	cpmadugsum.i	⊢ 𝐼 = ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑀 ) )
	cpmadugsum.j	⊢ 𝐽 = ( 𝑁 maAdju 𝑃 )
	cpmadugsum.0	⊢ 0 = ( 0g ‘ 𝑌 )
	cpmadugsum.g2	⊢ 𝐺 = ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) , if ( 𝑛 = ( 𝑠 + 1 ) , ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) , if ( ( 𝑠 + 1 ) < 𝑛 , 0 , ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) ) ) ) ) )
	cpmidgsum2.c	⊢ 𝐶 = ( 𝑁 CharPlyMat 𝑅 )
	cpmidgsum2.k	⊢ 𝐾 = ( 𝐶 ‘ 𝑀 )
	cpmidg2sum.u	⊢ 𝑈 = ( algSc ‘ 𝑃 )
Assertion	cpmidg2sum	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑠 ∈ ℕ ∃ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑈 ‘ ( ( coe1 ‘ 𝐾 ) ‘ 𝑖 ) ) · 1 ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cpmadugsum.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2		cpmadugsum.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
3		cpmadugsum.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
4		cpmadugsum.y	⊢ 𝑌 = ( 𝑁 Mat 𝑃 )
5		cpmadugsum.t	⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 )
6		cpmadugsum.x	⊢ 𝑋 = ( var1 ‘ 𝑅 )
7		cpmadugsum.e	⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑃 ) )
8		cpmadugsum.m	⊢ · = ( ·𝑠 ‘ 𝑌 )
9		cpmadugsum.r	⊢ × = ( .r ‘ 𝑌 )
10		cpmadugsum.1	⊢ 1 = ( 1r ‘ 𝑌 )
11		cpmadugsum.g	⊢ + = ( +g ‘ 𝑌 )
12		cpmadugsum.s	⊢ − = ( -g ‘ 𝑌 )
13		cpmadugsum.i	⊢ 𝐼 = ( ( 𝑋 · 1 ) − ( 𝑇 ‘ 𝑀 ) )
14		cpmadugsum.j	⊢ 𝐽 = ( 𝑁 maAdju 𝑃 )
15		cpmadugsum.0	⊢ 0 = ( 0g ‘ 𝑌 )
16		cpmadugsum.g2	⊢ 𝐺 = ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( 0 − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) , if ( 𝑛 = ( 𝑠 + 1 ) , ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) , if ( ( 𝑠 + 1 ) < 𝑛 , 0 , ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑛 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑛 ) ) ) ) ) ) ) )
17		cpmidgsum2.c	⊢ 𝐶 = ( 𝑁 CharPlyMat 𝑅 )
18		cpmidgsum2.k	⊢ 𝐾 = ( 𝐶 ‘ 𝑀 )
19		cpmidg2sum.u	⊢ 𝑈 = ( algSc ‘ 𝑃 )
20		eqid	⊢ ( 𝐾 · 1 ) = ( 𝐾 · 1 )
21	1 2 3 4 6 7 8 10 19 17 18 20	cpmidgsum	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝐾 · 1 ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑈 ‘ ( ( coe1 ‘ 𝐾 ) ‘ 𝑖 ) ) · 1 ) ) ) ) )
22	21	eqcomd	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑈 ‘ ( ( coe1 ‘ 𝐾 ) ‘ 𝑖 ) ) · 1 ) ) ) ) = ( 𝐾 · 1 ) )
23	22	ad3antrrr	⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ ( 𝐾 · 1 ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑈 ‘ ( ( coe1 ‘ 𝐾 ) ‘ 𝑖 ) ) · 1 ) ) ) ) = ( 𝐾 · 1 ) )
24		simpr	⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ ( 𝐾 · 1 ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) → ( 𝐾 · 1 ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )
25	23 24	eqtrd	⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ ( 𝐾 · 1 ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑈 ‘ ( ( coe1 ‘ 𝐾 ) ‘ 𝑖 ) ) · 1 ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )
26	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20	cpmidgsum2	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑠 ∈ ℕ ∃ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ( 𝐾 · 1 ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )
27	25 26	reximddv2	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑠 ∈ ℕ ∃ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑈 ‘ ( ( coe1 ‘ 𝐾 ) ‘ 𝑖 ) ) · 1 ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )