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Theorem cayleyhamiltonALT 21493

Description: Alternate proof of cayleyhamilton 21492, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 21491 directly, but has the same structure as the proof of cayleyhamilton0 21491. In contrast to the proof of cayleyhamilton0 21491, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
cayleyhamilton.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cayleyhamilton.b	⊢ 𝐵 = (Base‘𝐴)
cayleyhamilton.0	⊢ 0 = (0_g‘𝐴)
cayleyhamilton.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
cayleyhamilton.k	⊢ 𝐾 = (coe₁‘(𝐶‘𝑀))
cayleyhamilton.m	⊢ ∗ = ( ·_𝑠 ‘𝐴)
cayleyhamilton.e	⊢ ↑ = (._g‘(mulGrp‘𝐴))

**Assertion**
Ref	Expression
cayleyhamiltonALT	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 𝐶,𝑛 𝑛,𝑀 𝑛,𝑁 𝑅,𝑛 ∗ ,𝑛 ↑ ,𝑛 Allowed substitution hints: 𝐾(𝑛) 0 (𝑛)

Proof of Theorem cayleyhamiltonALT Dummy variables 𝑏 𝑚 𝑠 𝑥 𝑦 𝑙 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cayleyhamilton.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		cayleyhamilton.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
3		eqid 2821	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2821	. . . 4 ⊢ (𝑁 Mat (Poly₁‘𝑅)) = (𝑁 Mat (Poly₁‘𝑅))
5		eqid 2821	. . . 4 ⊢ (._r‘(𝑁 Mat (Poly₁‘𝑅))) = (._r‘(𝑁 Mat (Poly₁‘𝑅)))
6		eqid 2821	. . . 4 ⊢ (-_g‘(𝑁 Mat (Poly₁‘𝑅))) = (-_g‘(𝑁 Mat (Poly₁‘𝑅)))
7		eqid 2821	. . . 4 ⊢ (0_g‘(𝑁 Mat (Poly₁‘𝑅))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅)))
8		eqid 2821	. . . 4 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9		cayleyhamilton.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
10		eqid 2821	. . . 4 ⊢ (𝐶‘𝑀) = (𝐶‘𝑀)
11		eqeq1 2825	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12		eqeq1 2825	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13		breq2 5062	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14		oveq1 7157	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
15	14	fveq2d 6668	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
16	15	fveq2d 6668	. . . . . . . . 9 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑏‘𝑙) = (𝑏‘𝑛))
18	17	fveq2d 6668	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛)))
19	18	oveq2d 7166	. . . . . . . . 9 ⊢ (𝑙 = 𝑛 → (((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))) = (((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛))))
20	16, 19	oveq12d 7168	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙)))) = (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛)))))
21	13, 20	ifbieq2d 4491	. . . . . . 7 ⊢ (𝑙 = 𝑛 → if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))) = if((𝑠 + 1) < 𝑛, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛))))))
22	12, 21	ifbieq2d 4491	. . . . . 6 ⊢ (𝑙 = 𝑛 → if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙)))))) = if(𝑛 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛)))))))
23	11, 22	ifbieq2d 4491	. . . . 5 ⊢ (𝑙 = 𝑛 → if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))) = if(𝑛 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛))))))))
24	23	cbvmptv 5161	. . . 4 ⊢ (𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙)))))))) = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛))))))))
25		eqid 2821	. . . 4 ⊢ (Base‘(𝑁 Mat (Poly₁‘𝑅))) = (Base‘(𝑁 Mat (Poly₁‘𝑅)))
26		eqid 2821	. . . 4 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
27		cayleyhamilton.m	. . . 4 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
28		eqid 2821	. . . 4 ⊢ (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29		cayleyhamilton.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝐴))
30		eqid 2821	. . . 4 ⊢ (._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅)))) = (._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))
31	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 26, 27, 28, 29, 30	cayhamlem4 21490	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑁 cPolyMatToMat 𝑅)‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))))
32		eqid 2821	. . . . . . . . 9 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
33	28, 32	cpm2mfval 21351	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
34	33	eqcomd 2827	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35	34	3adant3 1128	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
36	35	fveq1d 6666	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) = ((𝑁 cPolyMatToMat 𝑅)‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))))
37	36	eqeq2d 2832	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) ↔ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑁 cPolyMatToMat 𝑅)‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))))))
38	37	2rexbidv 3300	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) ↔ ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑁 cPolyMatToMat 𝑅)‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))))))
39	31, 38	mpbird 259	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))))
40		cayleyhamilton.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (coe₁‘(𝐶‘𝑀))
41	40	eqcomi 2830	. . . . . . . . . . . 12 ⊢ (coe₁‘(𝐶‘𝑀)) = 𝐾
42	41	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 ∈ ℕ₀) → (coe₁‘(𝐶‘𝑀)) = 𝐾)
43	42	fveq1d 6666	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 ∈ ℕ₀) → ((coe₁‘(𝐶‘𝑀))‘𝑛) = (𝐾‘𝑛))
44	43	oveq1d 7165	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑛 ∈ ℕ₀) → (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)) = ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))
45	44	mpteq2dva 5153	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀))) = (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀))))
46	45	oveq2d 7166	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))))
47	46	eqeq1d 2823	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) ↔ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))))))
48	47	biimpa 479	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))))
49		oveq1 7157	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀)) = (𝑗(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀)))
50		fveq2 6664	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛) = ((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑗))
51	49, 50	oveq12d 7168	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)) = ((𝑗(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑗)))
52	51	cbvmptv 5161	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))) = (𝑗 ∈ ℕ₀ ↦ ((𝑗(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑗)))
53	52	oveq2i 7161	. . . . . . . . 9 ⊢ ((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))) = ((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑗 ∈ ℕ₀ ↦ ((𝑗(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑗))))
54	53	a1i 11	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))) = ((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑗 ∈ ℕ₀ ↦ ((𝑗(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑗)))))
55	1, 2, 3, 4, 5, 6, 7, 8, 24, 30	cayhamlem1 21468	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑗 ∈ ℕ₀ ↦ ((𝑗(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑗)))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅))))
56	54, 55	eqtrd 2856	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅))))
57		fveq2 6664	. . . . . . . 8 ⊢ (((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))))
58		crngring 19302	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
59	58	anim2i 618	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60	59	3adant3 1128	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
61	28, 32	cpm2mfval 21351	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
62	61	eqcomd 2827	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
63	62	fveq1d 6666	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = ((𝑁 cPolyMatToMat 𝑅)‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))))
64		eqid 2821	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
65	1, 28, 3, 4, 64, 7	m2cpminv0 21363	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = (0_g‘𝐴))
66	63, 65	eqtrd 2856	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = (0_g‘𝐴))
67	60, 66	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = (0_g‘𝐴))
68		cayleyhamilton.0	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝐴)
69	67, 68	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = 0 )
70	69	adantr 483	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = 0 )
71	57, 70	sylan9eqr 2878	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ ((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅)))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) = 0 )
72	56, 71	mpdan 685	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) = 0 )
73	72	adantr 483	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) = 0 )
74	48, 73	eqtrd 2856	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛)))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
75	74	ex 415	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))
76	75	rexlimdvva 3294	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly₁‘𝑅)) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛(._g‘(mulGrp‘(𝑁 Mat (Poly₁‘𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((0_g‘(𝑁 Mat (Poly₁‘𝑅)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑙, (0_g‘(𝑁 Mat (Poly₁‘𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-_g‘(𝑁 Mat (Poly₁‘𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(._r‘(𝑁 Mat (Poly₁‘𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙))))))))‘𝑛))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))
77	39, 76	mpd 15	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ifcif 4466 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ↑_m cmap 8400 Fincfn 8503 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 − cmin 10864 ℕcn 11632 ℕ₀cn0 11891 ...cfz 12886 Basecbs 16477 ._rcmulr 16560 ·_𝑠 cvsca 16563 0_gc0g 16707 Σ_g cgsu 16708 -_gcsg 18099 ._gcmg 18218 mulGrpcmgp 19233 1_rcur 19245 Ringcrg 19291 CRingccrg 19292 Poly₁cpl1 20339 coe₁cco1 20340 Mat cmat 21010 ConstPolyMat ccpmat 21305 matToPolyMat cmat2pmat 21306 cPolyMatToMat ccpmat2mat 21307 CharPlyMat cchpmat 21428

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1501 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-cur 7927 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-reverse 14115 df-s2 14204 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-efmnd 18028 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-gim 18393 df-cntz 18441 df-oppg 18468 df-symg 18490 df-pmtr 18564 df-psgn 18613 df-evpm 18614 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-srg 19250 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-rnghom 19461 df-drng 19498 df-subrg 19527 df-lmod 19630 df-lss 19698 df-sra 19938 df-rgmod 19939 df-assa 20079 df-ascl 20081 df-psr 20130 df-mvr 20131 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-vr1 20343 df-ply1 20344 df-coe1 20345 df-cnfld 20540 df-zring 20612 df-zrh 20645 df-dsmm 20870 df-frlm 20885 df-mamu 20989 df-mat 21011 df-mdet 21188 df-madu 21237 df-cpmat 21308 df-mat2pmat 21309 df-cpmat2mat 21310 df-decpmat 21365 df-pm2mp 21395 df-chpmat 21429

This theorem is referenced by: (None)

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