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Theorem cayleyhamiltonALT 23209

Description: Alternate proof of cayleyhamilton 23208, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 23207 directly, but has the same structure as the proof of cayleyhamilton0 23207. In contrast to the proof of cayleyhamilton0 23207, 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 2761	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2761	. . . 4 ⊢ (𝑁 Mat (Poly₁‘𝑅)) = (𝑁 Mat (Poly₁‘𝑅))
5		eqid 2761	. . . 4 ⊢ (._r‘(𝑁 Mat (Poly₁‘𝑅))) = (._r‘(𝑁 Mat (Poly₁‘𝑅)))
6		eqid 2761	. . . 4 ⊢ (-_g‘(𝑁 Mat (Poly₁‘𝑅))) = (-_g‘(𝑁 Mat (Poly₁‘𝑅)))
7		eqid 2761	. . . 4 ⊢ (0_g‘(𝑁 Mat (Poly₁‘𝑅))) = (0_g‘(𝑁 Mat (Poly₁‘𝑅)))
8		eqid 2761	. . . 4 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9		cayleyhamilton.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
10		eqid 2761	. . . 4 ⊢ (𝐶‘𝑀) = (𝐶‘𝑀)
11		eqeq1 2765	. . . . . 6 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12		eqeq1 2765	. . . . . . 7 ⊢ (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13		breq2 5107	. . . . . . . 8 ⊢ (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14		oveq1 7427	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
15	14	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
16	15	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (𝑏‘𝑙) = (𝑏‘𝑛))
18	17	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘𝑛)))
19	18	oveq2d 7436	. . . . . . . . 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 7438	. . . . . . . 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 4509	. . . . . . 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 4509	. . . . . 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 4509	. . . . 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 5209	. . . 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 2761	. . . 4 ⊢ (Base‘(𝑁 Mat (Poly₁‘𝑅))) = (Base‘(𝑁 Mat (Poly₁‘𝑅)))
26		eqid 2761	. . . 4 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
27		cayleyhamilton.m	. . . 4 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
28		eqid 2761	. . . 4 ⊢ (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29		cayleyhamilton.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝐴))
30		eqid 2761	. . . 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 23206	. . 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 2761	. . . . . . . . 9 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
33	28, 32	cpm2mfval 23067	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
34	33	eqcomd 2767	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35	34	3adant3 1150	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
36	35	fveq1d 6887	. . . . 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 2772	. . . 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 3228	. . 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 2770	. . . . . . . . . . . 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 6887	. . . . . . . . . 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 7435	. . . . . . . . 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 5198	. . . . . . . 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 7436	. . . . . . 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 2763	. . . . . 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 482	. . . . 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 7427	. . . . . . . . . . . 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 6885	. . . . . . . . . . . 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 7438	. . . . . . . . . . 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 5209	. . . . . . . . . 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 7431	. . . . . . . . 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 23184	. . . . . . . 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 2796	. . . . . . 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 6885	. . . . . . . 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 20472	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
59	58	anim2i 629	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60	59	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
61	28, 32	cpm2mfval 23067	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
62	61	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
63	62	fveq1d 6887	. . . . . . . . . . . 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 2761	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
65	1, 28, 3, 4, 64, 7	m2cpminv0 23079	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0_g‘(𝑁 Mat (Poly₁‘𝑅)))) = (0_g‘𝐴))
66	63, 65	eqtrd 2796	. . . . . . . . . . 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 2814	. . . . . . . . 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 486	. . . . . . . 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 2818	. . . . . . 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 700	. . . . . 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 486	. . . . 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 2796	. . . 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 418	. . 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 3220	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∃wrex 3087 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 ↑_m cmap 8847 Fincfn 8973 0cc0 11200 1c1 11201 + caddc 11203 < clt 11343 − cmin 11541 ℕcn 12335 ℕ₀cn0 12606 ...cfz 13639 Basecbs 17387 ._rcmulr 17429 ·_𝑠 cvsca 17432 0_gc0g 17610 Σ_g cgsu 17611 -_gcsg 19146 ._gcmg 19277 mulGrpcmgp 20360 1_rcur 20407 Ringcrg 20459 CRingccrg 20460 Poly₁cpl1 22495 coe₁cco1 22496 Mat cmat 22722 ConstPolyMat ccpmat 23021 matToPolyMat cmat2pmat 23022 cPolyMatToMat ccpmat2mat 23023 CharPlyMat cchpmat 23144

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-addf 11279 ax-mulf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-cur 8284 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-word 14659 df-lsw 14708 df-concat 14716 df-s1 14743 df-substr 14789 df-pfx 14821 df-splice 14899 df-reverse 14908 df-s2 14999 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-0g 17612 df-gsum 17613 df-prds 17618 df-pws 17620 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-efmnd 19065 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-ghm 19428 df-gim 19473 df-cntz 19531 df-oppg 19560 df-symg 19584 df-pmtr 19656 df-psgn 19705 df-evpm 19706 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-srg 20413 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-rhm 20702 df-subrng 20798 df-subrg 20822 df-drng 20982 df-lmod 21137 df-lss 21207 df-sra 21448 df-rgmod 21449 df-cnfld 21679 df-zring 21753 df-zrh 21809 df-dsmm 22038 df-frlm 22053 df-assa 22161 df-ascl 22163 df-psr 22217 df-mvr 22218 df-mpl 22219 df-opsr 22221 df-psr1 22498 df-vr1 22499 df-ply1 22500 df-coe1 22501 df-mamu 22706 df-mat 22723 df-mdet 22900 df-madu 22949 df-cpmat 23024 df-mat2pmat 23025 df-cpmat2mat 23026 df-decpmat 23081 df-pm2mp 23111 df-chpmat 23145

This theorem is used by: (None)

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