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Theorem cayleyhamiltonALT 22780

Description: Alternate proof of cayleyhamilton 22779, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22778 directly, but has the same structure as the proof of cayleyhamilton0 22778. In contrast to the proof of cayleyhamilton0 22778, 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 2727	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2727	. . . 4 ⊢ (𝑁 Mat (Poly₁‘𝑅)) = (𝑁 Mat (Poly₁‘𝑅))
5		eqid 2727	. . . 4 ⊢ (._r‘(𝑁 Mat (Poly₁‘𝑅))) = (._r‘(𝑁 Mat (Poly₁‘𝑅)))
6		eqid 2727	. . . 4 ⊢ (-_g‘(𝑁 Mat (Poly₁‘𝑅))) = (-_g‘(𝑁 Mat (Poly₁‘𝑅)))
7		eqid 2727	. . . 4 ⊢ (0_g‘(𝑁 Mat (Poly₁‘𝑅))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅)))
8		eqid 2727	. . . 4 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9		cayleyhamilton.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
10		eqid 2727	. . . 4 ⊢ (𝐶‘𝑀) = (𝐶‘𝑀)
11		eqeq1 2731	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12		eqeq1 2731	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13		breq2 5146	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14		oveq1 7421	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
15	14	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
16	15	fveq2d 6895	. . . . . . . . 9 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑏‘𝑙) = (𝑏‘𝑛))
18	17	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛)))
19	18	oveq2d 7430	. . . . . . . . 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 7432	. . . . . . . 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 4550	. . . . . . 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 4550	. . . . . 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 4550	. . . . 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 5255	. . . 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 2727	. . . 4 ⊢ (Base‘(𝑁 Mat (Poly₁‘𝑅))) = (Base‘(𝑁 Mat (Poly₁‘𝑅)))
26		eqid 2727	. . . 4 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
27		cayleyhamilton.m	. . . 4 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
28		eqid 2727	. . . 4 ⊢ (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29		cayleyhamilton.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝐴))
30		eqid 2727	. . . 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 22777	. . 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 2727	. . . . . . . . 9 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
33	28, 32	cpm2mfval 22638	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
34	33	eqcomd 2733	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35	34	3adant3 1130	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
36	35	fveq1d 6893	. . . . 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 2738	. . . 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 3214	. . 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 257	. 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 2736	. . . . . . . . . . . 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 6893	. . . . . . . . . 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 7429	. . . . . . . . 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 5242	. . . . . . . 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 7430	. . . . . . 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 2729	. . . . . 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 476	. . . . 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 7421	. . . . . . . . . . . 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 6891	. . . . . . . . . . . 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 7432	. . . . . . . . . . 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 5255	. . . . . . . . . 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 7425	. . . . . . . . 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 22755	. . . . . . . 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 2767	. . . . . . 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 6891	. . . . . . . 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 20176	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
59	58	anim2i 616	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60	59	3adant3 1130	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
61	28, 32	cpm2mfval 22638	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
62	61	eqcomd 2733	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
63	62	fveq1d 6893	. . . . . . . . . . . 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 2727	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
65	1, 28, 3, 4, 64, 7	m2cpminv0 22650	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = (0_g‘𝐴))
66	63, 65	eqtrd 2767	. . . . . . . . . . 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	eqtr4di 2785	. . . . . . . . 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 480	. . . . . . . 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 2789	. . . . . . 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 686	. . . . . 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 480	. . . . 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 2767	. . . 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 412	. . 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 3206	. 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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 ifcif 4524 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ↑_m cmap 8836 Fincfn 8955 0cc0 11130 1c1 11131 + caddc 11133 < clt 11270 − cmin 11466 ℕcn 12234 ℕ₀cn0 12494 ...cfz 13508 Basecbs 17171 ._rcmulr 17225 ·_𝑠 cvsca 17228 0_gc0g 17412 Σ_g cgsu 17413 -_gcsg 18883 ._gcmg 19014 mulGrpcmgp 20065 1_rcur 20112 Ringcrg 20164 CRingccrg 20165 Poly₁cpl1 22083 coe₁cco1 22084 Mat cmat 22294 ConstPolyMat ccpmat 22592 matToPolyMat cmat2pmat 22593 cPolyMatToMat ccpmat2mat 22594 CharPlyMat cchpmat 22715

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-addf 11209 ax-mulf 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-cur 8266 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-xnn0 12567 df-z 12581 df-dec 12700 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-word 14489 df-lsw 14537 df-concat 14545 df-s1 14570 df-substr 14615 df-pfx 14645 df-splice 14724 df-reverse 14733 df-s2 14823 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-efmnd 18812 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-gim 19204 df-cntz 19259 df-oppg 19288 df-symg 19313 df-pmtr 19388 df-psgn 19437 df-evpm 19438 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-srg 20118 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-drng 20615 df-lmod 20734 df-lss 20805 df-sra 21047 df-rgmod 21048 df-cnfld 21267 df-zring 21360 df-zrh 21416 df-dsmm 21653 df-frlm 21668 df-assa 21774 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-coe1 22089 df-mamu 22273 df-mat 22295 df-mdet 22474 df-madu 22523 df-cpmat 22595 df-mat2pmat 22596 df-cpmat2mat 22597 df-decpmat 22652 df-pm2mp 22682 df-chpmat 22716

This theorem is referenced by: (None)

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