cayleyhamiltonALT - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cayleyhamiltonALT		Structured version Visualization version GIF version

Theorem cayleyhamiltonALT 22834

Description: Alternate proof of cayleyhamilton 22833, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22832 directly, but has the same structure as the proof of cayleyhamilton0 22832. In contrast to the proof of cayleyhamilton0 22832, 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 2736	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2736	. . . 4 ⊢ (𝑁 Mat (Poly₁‘𝑅)) = (𝑁 Mat (Poly₁‘𝑅))
5		eqid 2736	. . . 4 ⊢ (._r‘(𝑁 Mat (Poly₁‘𝑅))) = (._r‘(𝑁 Mat (Poly₁‘𝑅)))
6		eqid 2736	. . . 4 ⊢ (-_g‘(𝑁 Mat (Poly₁‘𝑅))) = (-_g‘(𝑁 Mat (Poly₁‘𝑅)))
7		eqid 2736	. . . 4 ⊢ (0_g‘(𝑁 Mat (Poly₁‘𝑅))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅)))
8		eqid 2736	. . . 4 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9		cayleyhamilton.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
10		eqid 2736	. . . 4 ⊢ (𝐶‘𝑀) = (𝐶‘𝑀)
11		eqeq1 2740	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12		eqeq1 2740	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13		breq2 5128	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14		oveq1 7417	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
15	14	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
16	15	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑏‘𝑙) = (𝑏‘𝑛))
18	17	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛)))
19	18	oveq2d 7426	. . . . . . . . 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 7428	. . . . . . . 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 4532	. . . . . . 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 4532	. . . . . 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 4532	. . . . 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 5230	. . . 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 2736	. . . 4 ⊢ (Base‘(𝑁 Mat (Poly₁‘𝑅))) = (Base‘(𝑁 Mat (Poly₁‘𝑅)))
26		eqid 2736	. . . 4 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
27		cayleyhamilton.m	. . . 4 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
28		eqid 2736	. . . 4 ⊢ (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29		cayleyhamilton.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝐴))
30		eqid 2736	. . . 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 22831	. . 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 2736	. . . . . . . . 9 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
33	28, 32	cpm2mfval 22692	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
34	33	eqcomd 2742	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35	34	3adant3 1132	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
36	35	fveq1d 6883	. . . . 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 2747	. . . 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 3210	. . 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 2745	. . . . . . . . . . . 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 6883	. . . . . . . . . 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 7425	. . . . . . . . 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 5219	. . . . . . . 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 7426	. . . . . . 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 2738	. . . . . 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 7417	. . . . . . . . . . . 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 6881	. . . . . . . . . . . 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 7428	. . . . . . . . . . 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 5230	. . . . . . . . . 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 7421	. . . . . . . . 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 22809	. . . . . . . 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 2771	. . . . . . 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 6881	. . . . . . . 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 20210	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
59	58	anim2i 617	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60	59	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
61	28, 32	cpm2mfval 22692	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
62	61	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
63	62	fveq1d 6883	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
65	1, 28, 3, 4, 64, 7	m2cpminv0 22704	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = (0_g‘𝐴))
66	63, 65	eqtrd 2771	. . . . . . . . . . 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 2789	. . . . . . . . 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 2793	. . . . . . 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 687	. . . . . 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 2771	. . . 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 3202	. 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 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3061 ifcif 4505 class class class wbr 5124 ↦ cmpt 5206 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 ↑_m cmap 8845 Fincfn 8964 0cc0 11134 1c1 11135 + caddc 11137 < clt 11274 − cmin 11471 ℕcn 12245 ℕ₀cn0 12506 ...cfz 13529 Basecbs 17233 ._rcmulr 17277 ·_𝑠 cvsca 17280 0_gc0g 17458 Σ_g cgsu 17459 -_gcsg 18923 ._gcmg 19055 mulGrpcmgp 20105 1_rcur 20146 Ringcrg 20198 CRingccrg 20199 Poly₁cpl1 22117 coe₁cco1 22118 Mat cmat 22350 ConstPolyMat ccpmat 22646 matToPolyMat cmat2pmat 22647 cPolyMatToMat ccpmat2mat 22648 CharPlyMat cchpmat 22769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-addf 11213 ax-mulf 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-ot 4615 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 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 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-cur 8271 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-xnn0 12580 df-z 12594 df-dec 12714 df-uz 12858 df-rp 13014 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-hash 14354 df-word 14537 df-lsw 14586 df-concat 14594 df-s1 14619 df-substr 14664 df-pfx 14694 df-splice 14773 df-reverse 14782 df-s2 14872 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-efmnd 18852 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-gim 19247 df-cntz 19305 df-oppg 19334 df-symg 19356 df-pmtr 19428 df-psgn 19477 df-evpm 19478 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-srg 20152 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-subrng 20511 df-subrg 20535 df-drng 20696 df-lmod 20824 df-lss 20894 df-sra 21136 df-rgmod 21137 df-cnfld 21321 df-zring 21413 df-zrh 21469 df-dsmm 21697 df-frlm 21712 df-assa 21818 df-ascl 21820 df-psr 21874 df-mvr 21875 df-mpl 21876 df-opsr 21878 df-psr1 22120 df-vr1 22121 df-ply1 22122 df-coe1 22123 df-mamu 22334 df-mat 22351 df-mdet 22528 df-madu 22577 df-cpmat 22649 df-mat2pmat 22650 df-cpmat2mat 22651 df-decpmat 22706 df-pm2mp 22736 df-chpmat 22770

This theorem is referenced by: (None)

W3C validator