Theorem cpmadugsum

Description: The product of the characteristic matrix of a given matrix and its adjunct represented as an infinite sum. (Contributed by AV, 10-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 ) ) ) ) ) ) ) )
Assertion	cpmadugsum	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑠 ∈ ℕ ∃ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( 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	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cpmadugsumfi	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑠 ∈ ℕ ∃ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) )
18		simpr	⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) → ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) )
19	1 2 3 4 9 12 15 5 16 6 8 7 11	chfacfscmulgsum	⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) )
20	19	eqcomd	⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )
21	20	adantr	⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) → ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )
22	18 21	eqtrd	⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) ∧ ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) ) → ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )
23	22	ex	⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → ( ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) → ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) )
24	23	reximdvva	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( ∃ 𝑠 ∈ ℕ ∃ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( ( 𝑌 Σg ( 𝑖 ∈ ( 1 ... 𝑠 ) ↦ ( ( 𝑖 ↑ 𝑋 ) · ( ( 𝑇 ‘ ( 𝑏 ‘ ( 𝑖 − 1 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 𝑖 ) ) ) ) ) ) ) + ( ( ( ( 𝑠 + 1 ) ↑ 𝑋 ) · ( 𝑇 ‘ ( 𝑏 ‘ 𝑠 ) ) ) − ( ( 𝑇 ‘ 𝑀 ) × ( 𝑇 ‘ ( 𝑏 ‘ 0 ) ) ) ) ) → ∃ 𝑠 ∈ ℕ ∃ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) ) )
25	17 24	mpd	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑠 ∈ ℕ ∃ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ( 𝐼 × ( 𝐽 ‘ 𝐼 ) ) = ( 𝑌 Σg ( 𝑖 ∈ ℕ0 ↦ ( ( 𝑖 ↑ 𝑋 ) · ( 𝐺 ‘ 𝑖 ) ) ) ) )