Theorem cayleyhamilton1 22830

Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 22828, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹‘𝑛), then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)

Hypotheses

Ref	Expression
cayleyhamilton.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cayleyhamilton.b	⊢ 𝐵 = (Base‘𝐴)
cayleyhamilton.0	⊢ 0 = (0g‘𝐴)
cayleyhamilton.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
cayleyhamilton.k	⊢ 𝐾 = (coe1‘(𝐶‘𝑀))
cayleyhamilton.m	⊢ ∗ = ( ·𝑠 ‘𝐴)
cayleyhamilton.e	⊢ ↑ = (.g‘(mulGrp‘𝐴))
cayleyhamilton1.l	⊢ 𝐿 = (Base‘𝑅)
cayleyhamilton1.x	⊢ 𝑋 = (var1‘𝑅)
cayleyhamilton1.p	⊢ 𝑃 = (Poly1‘𝑅)
cayleyhamilton1.m	⊢ · = ( ·𝑠 ‘𝑃)
cayleyhamilton1.e	⊢ 𝐸 = (.g‘(mulGrp‘𝑃))
cayleyhamilton1.z	⊢ 𝑍 = (0g‘𝑅)

Assertion

Ref	Expression
cayleyhamilton1	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝐶,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   ∗ ,𝑛   ↑ ,𝑛   𝑛,𝐸   𝑛,𝐹   𝑛,𝐿   𝑃,𝑛   𝑛,𝑋   𝑛,𝑍   · ,𝑛

Allowed substitution hints:   𝐾(𝑛)   0 (𝑛)

Proof of Theorem cayleyhamilton1

Dummy variables 𝑚 𝑖 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		cayleyhamilton.0	. . . 4 ⊢ 0 = (0g‘𝐴)
4		cayleyhamilton.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
5		cayleyhamilton.k	. . . 4 ⊢ 𝐾 = (coe1‘(𝐶‘𝑀))
6		cayleyhamilton.m	. . . 4 ⊢ ∗ = ( ·𝑠 ‘𝐴)
7		cayleyhamilton.e	. . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝐴))
8	1, 2, 3, 4, 5, 6, 7	cayleyhamilton 22828	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
9	8	adantr 480	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
10		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))
11		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑃
12		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑛 Σg
13		nfmpt1 5220	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
14	11, 12, 13	nfov 7435	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
15	14	nfeq2 2916	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
16	10, 15	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑛(((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))))
17		crngring 20205	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18	17	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
19	18	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → 𝑅 ∈ Ring)
20		cayleyhamilton1.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
21		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘𝑃)
22	4, 1, 2, 20, 21	chpmatply1 22770	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
23	22	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝐶‘𝑀) ∈ (Base‘𝑃))
24		cayleyhamilton1.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝑅)
25		cayleyhamilton1.e	. . . . . . . . . . . 12 ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))
26		cayleyhamilton1.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (Base‘𝑅)
27		cayleyhamilton1.m	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑃)
28		eqid 2735	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
29		elmapi 8863	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹:ℕ0⟶𝐿)
30		ffvelcdm 7071	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑛 ∈ ℕ0) → (𝐹‘𝑛) ∈ 𝐿)
31	30	ralrimiva 3132	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ0⟶𝐿 → ∀𝑛 ∈ ℕ0 (𝐹‘𝑛) ∈ 𝐿)
32	29, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → ∀𝑛 ∈ ℕ0 (𝐹‘𝑛) ∈ 𝐿)
33	32	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑛 ∈ ℕ0 (𝐹‘𝑛) ∈ 𝐿)
34	29	feqmptd 6947	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝐹‘𝑛)))
35		cayleyhamilton1.z	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 = (0g‘𝑅)
36	35	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝑍 = (0g‘𝑅))
37	34, 36	breq12d 5132	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → (𝐹 finSupp 𝑍 ↔ (𝑛 ∈ ℕ0 ↦ (𝐹‘𝑛)) finSupp (0g‘𝑅)))
38	37	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍) → (𝑛 ∈ ℕ0 ↦ (𝐹‘𝑛)) finSupp (0g‘𝑅))
39	38	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝑛 ∈ ℕ0 ↦ (𝐹‘𝑛)) finSupp (0g‘𝑅))
40	20, 21, 24, 25, 19, 26, 27, 28, 33, 39	gsumsmonply1 22245	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) ∈ (Base‘𝑃))
41		fveq2 6876	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
42		oveq1 7412	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
43	41, 42	oveq12d 7423	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → ((𝐹‘𝑖) · (𝑖𝐸𝑋)) = ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
44	43	cbvmptv 5225	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
45	44	oveq2i 7416	. . . . . . . . . . . . 13 ⊢ (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
46	45	fveq2i 6879	. . . . . . . . . . . 12 ⊢ (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))))
47	20, 21, 5, 46	ply1coe1eq 22238	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝐶‘𝑀) ∈ (Base‘𝑃) ∧ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) ∈ (Base‘𝑃)) → (∀𝑚 ∈ ℕ0 (𝐾‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))))
48	19, 23, 40, 47	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (∀𝑚 ∈ ℕ0 (𝐾‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))))
49		fveq2 6876	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝐾‘𝑚) = (𝐾‘𝑛))
50		fveq2 6876	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
51	49, 50	eqeq12d 2751	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → ((𝐾‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛)))
52	51	rspcva 3599	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℕ0 (𝐾‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚)) → (𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
53		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) ∧ (𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)))) → (𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
54	18	ad2antrl 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → 𝑅 ∈ Ring)
55		ffvelcdm 7071	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑖 ∈ ℕ0) → (𝐹‘𝑖) ∈ 𝐿)
56	55	ralrimiva 3132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:ℕ0⟶𝐿 → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
57	29, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
58	57	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
59	58	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
60	29	feqmptd 6947	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)))
61	60	breq1d 5129	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → (𝐹 finSupp 𝑍 ↔ (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍))
62	61	biimpa 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
63	62	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
64	63	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
65		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → 𝑛 ∈ ℕ0)
66	20, 21, 24, 25, 54, 26, 27, 35, 59, 64, 65	gsummoncoe1 22246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = ⦋𝑛 / 𝑖⦌(𝐹‘𝑖))
67		csbfv 6926	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋𝑛 / 𝑖⦌(𝐹‘𝑖) = (𝐹‘𝑛)
68	66, 67	eqtrdi 2786	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = (𝐹‘𝑛))
69	68	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) ∧ (𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = (𝐹‘𝑛))
70	53, 69	eqtrd 2770	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) ∧ (𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)))) → (𝐾‘𝑛) = (𝐹‘𝑛))
71	70	exp32 420	. . . . . . . . . . . . . . 15 ⊢ ((𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) → (𝑛 ∈ ℕ0 → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝐾‘𝑛) = (𝐹‘𝑛))))
72	71	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → ((𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝐾‘𝑛) = (𝐹‘𝑛))))
73	72	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℕ0 (𝐾‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚)) → ((𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝐾‘𝑛) = (𝐹‘𝑛))))
74	52, 73	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℕ0 (𝐾‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚)) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝐾‘𝑛) = (𝐹‘𝑛)))
75	74	com12 32	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℕ0 (𝐾‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚)) → (𝐾‘𝑛) = (𝐹‘𝑛)))
76	75	expcomd 416	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (∀𝑚 ∈ ℕ0 (𝐾‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑚) → (𝑛 ∈ ℕ0 → (𝐾‘𝑛) = (𝐹‘𝑛))))
77	48, 76	sylbird 260	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) → (𝑛 ∈ ℕ0 → (𝐾‘𝑛) = (𝐹‘𝑛))))
78	77	imp31 417	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))) ∧ 𝑛 ∈ ℕ0) → (𝐾‘𝑛) = (𝐹‘𝑛))
79	78	oveq1d 7420	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))) ∧ 𝑛 ∈ ℕ0) → ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)) = ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))
80	16, 79	mpteq2da 5213	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))) → (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀))) = (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀))))
81	80	oveq2d 7421	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))))
82	81	eqeq1d 2737	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ↔ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))
83	82	biimpd 229	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))
84	83	ex 412	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )))
85	9, 84	mpid 44	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ⦋csb 3874   class class class wbr 5119   ↦ cmpt 5201  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405   ↑_m cmap 8840  Fincfn 8959   finSupp cfsupp 9373  ℕ₀cn0 12501  Basecbs 17228   ·_𝑠 cvsca 17275  0_gc0g 17453   Σ_g cgsu 17454  ._gcmg 19050  mulGrpcmgp 20100  Ringcrg 20193  CRingccrg 20194  var₁cv1 22111  Poly₁cpl1 22112  coe₁cco1 22113   Mat cmat 22345   CharPlyMat cchpmat 22764

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-addf 11208  ax-mulf 11209

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-ofr 7672  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-tpos 8225  df-cur 8266  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-sup 9454  df-oi 9524  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-xnn0 12575  df-z 12589  df-dec 12709  df-uz 12853  df-rp 13009  df-fz 13525  df-fzo 13672  df-seq 14020  df-exp 14080  df-hash 14349  df-word 14532  df-lsw 14581  df-concat 14589  df-s1 14614  df-substr 14659  df-pfx 14689  df-splice 14768  df-reverse 14777  df-s2 14867  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-starv 17286  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-unif 17294  df-hom 17295  df-cco 17296  df-0g 17455  df-gsum 17456  df-prds 17461  df-pws 17463  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-efmnd 18847  df-grp 18919  df-minusg 18920  df-sbg 18921  df-mulg 19051  df-subg 19106  df-ghm 19196  df-gim 19242  df-cntz 19300  df-oppg 19329  df-symg 19351  df-pmtr 19423  df-psgn 19472  df-evpm 19473  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-srg 20147  df-ring 20195  df-cring 20196  df-oppr 20297  df-dvdsr 20317  df-unit 20318  df-invr 20348  df-dvr 20361  df-rhm 20432  df-subrng 20506  df-subrg 20530  df-drng 20691  df-lmod 20819  df-lss 20889  df-sra 21131  df-rgmod 21132  df-cnfld 21316  df-zring 21408  df-zrh 21464  df-dsmm 21692  df-frlm 21707  df-assa 21813  df-ascl 21815  df-psr 21869  df-mvr 21870  df-mpl 21871  df-opsr 21873  df-psr1 22115  df-vr1 22116  df-ply1 22117  df-coe1 22118  df-mamu 22329  df-mat 22346  df-mdet 22523  df-madu 22572  df-cpmat 22644  df-mat2pmat 22645  df-cpmat2mat 22646  df-decpmat 22701  df-pm2mp 22731  df-chpmat 22765

This theorem is referenced by: (None)