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Theorem cayleyhamiltonALT 23013

Description: Alternate proof of cayleyhamilton 23012, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 23011 directly, but has the same structure as the proof of cayleyhamilton0 23011. In contrast to the proof of cayleyhamilton0 23011, 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 2769	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2769	. . . 4 ⊢ (𝑁 Mat (Poly₁‘𝑅)) = (𝑁 Mat (Poly₁‘𝑅))
5		eqid 2769	. . . 4 ⊢ (._r‘(𝑁 Mat (Poly₁‘𝑅))) = (._r‘(𝑁 Mat (Poly₁‘𝑅)))
6		eqid 2769	. . . 4 ⊢ (-_g‘(𝑁 Mat (Poly₁‘𝑅))) = (-_g‘(𝑁 Mat (Poly₁‘𝑅)))
7		eqid 2769	. . . 4 ⊢ (0_g‘(𝑁 Mat (Poly₁‘𝑅))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅)))
8		eqid 2769	. . . 4 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9		cayleyhamilton.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
10		eqid 2769	. . . 4 ⊢ (𝐶‘𝑀) = (𝐶‘𝑀)
11		eqeq1 2773	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12		eqeq1 2773	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13		breq2 5114	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14		oveq1 7415	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
15	14	fveq2d 6883	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
16	15	fveq2d 6883	. . . . . . . . 9 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17		fveq2 6879	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑏‘𝑙) = (𝑏‘𝑛))
18	17	fveq2d 6883	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛)))
19	18	oveq2d 7424	. . . . . . . . 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 7426	. . . . . . . 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 4516	. . . . . . 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 4516	. . . . . 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 4516	. . . . 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 5216	. . . 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 2769	. . . 4 ⊢ (Base‘(𝑁 Mat (Poly₁‘𝑅))) = (Base‘(𝑁 Mat (Poly₁‘𝑅)))
26		eqid 2769	. . . 4 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
27		cayleyhamilton.m	. . . 4 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
28		eqid 2769	. . . 4 ⊢ (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29		cayleyhamilton.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝐴))
30		eqid 2769	. . . 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 23010	. . 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 2769	. . . . . . . . 9 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
33	28, 32	cpm2mfval 22871	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
34	33	eqcomd 2775	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35	34	3adant3 1148	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
36	35	fveq1d 6881	. . . . 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 2780	. . . 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 3236	. . 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 260	. 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 2778	. . . . . . . . . . . 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 6881	. . . . . . . . . 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 7423	. . . . . . . . 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 5205	. . . . . . . 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 7424	. . . . . . 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 2771	. . . . . 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 481	. . . . 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 7415	. . . . . . . . . . . 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 6879	. . . . . . . . . . . 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 7426	. . . . . . . . . . 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 5216	. . . . . . . . . 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 7419	. . . . . . . . 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 22988	. . . . . . . 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 2804	. . . . . . 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 6879	. . . . . . . 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 20323	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
59	58	anim2i 628	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60	59	3adant3 1148	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
61	28, 32	cpm2mfval 22871	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
62	61	eqcomd 2775	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
63	62	fveq1d 6881	. . . . . . . . . . . 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 2769	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
65	1, 28, 3, 4, 64, 7	m2cpminv0 22883	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = (0_g‘𝐴))
66	63, 65	eqtrd 2804	. . . . . . . . . . 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 18	. . . . . . . . . 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 2822	. . . . . . . . 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 485	. . . . . . . 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 2826	. . . . . . 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 699	. . . . . 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 485	. . . . 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 2804	. . . 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 417	. . 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 3228	. 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 16	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 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∃wrex 3095 ifcif 4489 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6534 (class class class)co 7408 ∈ cmpo 7410 ↑_m cmap 8820 Fincfn 8939 0cc0 11096 1c1 11097 + caddc 11099 < clt 11239 − cmin 11437 ℕcn 12229 ℕ₀cn0 12500 ...cfz 13531 Basecbs 17265 ._rcmulr 17307 ·_𝑠 cvsca 17310 0_gc0g 17488 Σ_g cgsu 17489 -_gcsg 18998 ._gcmg 19129 mulGrpcmgp 20212 1_rcur 20259 Ringcrg 20311 CRingccrg 20312 Poly₁cpl1 22302 coe₁cco1 22303 Mat cmat 22529 ConstPolyMat ccpmat 22825 matToPolyMat cmat2pmat 22826 cPolyMatToMat ccpmat2mat 22827 CharPlyMat cchpmat 22948

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-addf 11175 ax-mulf 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-xor 1539 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-ofr 7673 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-cur 8259 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-oi 9468 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-xnn0 12574 df-z 12588 df-dec 12708 df-uz 12859 df-rp 13013 df-fz 13532 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-word 14547 df-lsw 14596 df-concat 14604 df-s1 14630 df-substr 14675 df-pfx 14705 df-splice 14783 df-reverse 14792 df-s2 14881 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-0g 17490 df-gsum 17491 df-prds 17496 df-pws 17498 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-mhm 18837 df-submnd 18838 df-efmnd 18924 df-grp 18999 df-minusg 19000 df-sbg 19001 df-mulg 19130 df-subg 19185 df-ghm 19280 df-gim 19325 df-cntz 19383 df-oppg 19412 df-symg 19436 df-pmtr 19508 df-psgn 19557 df-evpm 19558 df-cmn 19848 df-abl 19849 df-mgp 20213 df-rng 20227 df-ur 20260 df-srg 20265 df-ring 20313 df-cring 20314 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-rhm 20550 df-subrng 20627 df-subrg 20651 df-drng 20811 df-lmod 20957 df-lss 21027 df-sra 21268 df-rgmod 21269 df-cnfld 21488 df-zring 21562 df-zrh 21618 df-dsmm 21847 df-frlm 21862 df-assa 21968 df-ascl 21970 df-psr 22024 df-mvr 22025 df-mpl 22026 df-opsr 22028 df-psr1 22305 df-vr1 22306 df-ply1 22307 df-coe1 22308 df-mamu 22513 df-mat 22530 df-mdet 22707 df-madu 22756 df-cpmat 22828 df-mat2pmat 22829 df-cpmat2mat 22830 df-decpmat 22885 df-pm2mp 22915 df-chpmat 22949

This theorem is referenced by: (None)

W3C validator