Theorem cayleyhamilton1 22807

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 22805, 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 22805	. . 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 1915	. . . . . . . 8 ⊢ Ⅎ𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))
11		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑃
12		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑛 Σg
13		nfmpt1 5188	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
14	11, 12, 13	nfov 7376	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
15	14	nfeq2 2912	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
16	10, 15	nfan 1900	. . . . . . 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 20163	. . . . . . . . . . . . 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 2731	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘𝑃)
22	4, 1, 2, 20, 21	chpmatply1 22747	. . . . . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
29		elmapi 8773	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹:ℕ0⟶𝐿)
30		ffvelcdm 7014	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑛 ∈ ℕ0) → (𝐹‘𝑛) ∈ 𝐿)
31	30	ralrimiva 3124	. . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . 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 22222	. . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
42		oveq1 7353	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
43	41, 42	oveq12d 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → ((𝐹‘𝑖) · (𝑖𝐸𝑋)) = ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
44	43	cbvmptv 5193	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
45	44	oveq2i 7357	. . . . . . . . . . . . 13 ⊢ (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
46	45	fveq2i 6825	. . . . . . . . . . . 12 ⊢ (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))))
47	20, 21, 5, 46	ply1coe1eq 22215	. . . . . . . . . . 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 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝐾‘𝑚) = (𝐾‘𝑛))
50		fveq2 6822	. . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . 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 3570	. . . . . . . . . . . . 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 7014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑖 ∈ ℕ0) → (𝐹‘𝑖) ∈ 𝐿)
56	55	ralrimiva 3124	. . . . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)))
61	60	breq1d 5099	. . . . . . . . . . . . . . . . . . . . . . 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 22223	. . . . . . . . . . . . . . . . . . 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 6869	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋𝑛 / 𝑖⦌(𝐹‘𝑖) = (𝐹‘𝑛)
68	66, 67	eqtrdi 2782	. . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 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 7361	. . . . . . 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 5181	. . . . . 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 7362	. . . . 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 2733	. . . 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 1541   ∈ wcel 2111  ∀wral 3047  ⦋csb 3845   class class class wbr 5089   ↦ cmpt 5170  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  Fincfn 8869   finSupp cfsupp 9245  ℕ₀cn0 12381  Basecbs 17120   ·_𝑠 cvsca 17165  0_gc0g 17343   Σ_g cgsu 17344  ._gcmg 18980  mulGrpcmgp 20058  Ringcrg 20151  CRingccrg 20152  var₁cv1 22088  Poly₁cpl1 22089  coe₁cco1 22090   Mat cmat 22322   CharPlyMat cchpmat 22741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-addf 11085  ax-mulf 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-ofr 7611  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-tpos 8156  df-cur 8197  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-sup 9326  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-xnn0 12455  df-z 12469  df-dec 12589  df-uz 12733  df-rp 12891  df-fz 13408  df-fzo 13555  df-seq 13909  df-exp 13969  df-hash 14238  df-word 14421  df-lsw 14470  df-concat 14478  df-s1 14504  df-substr 14549  df-pfx 14579  df-splice 14657  df-reverse 14666  df-s2 14755  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-submnd 18692  df-efmnd 18777  df-grp 18849  df-minusg 18850  df-sbg 18851  df-mulg 18981  df-subg 19036  df-ghm 19125  df-gim 19171  df-cntz 19229  df-oppg 19258  df-symg 19282  df-pmtr 19354  df-psgn 19403  df-evpm 19404  df-cmn 19694  df-abl 19695  df-mgp 20059  df-rng 20071  df-ur 20100  df-srg 20105  df-ring 20153  df-cring 20154  df-oppr 20255  df-dvdsr 20275  df-unit 20276  df-invr 20306  df-dvr 20319  df-rhm 20390  df-subrng 20461  df-subrg 20485  df-drng 20646  df-lmod 20795  df-lss 20865  df-sra 21107  df-rgmod 21108  df-cnfld 21292  df-zring 21384  df-zrh 21440  df-dsmm 21669  df-frlm 21684  df-assa 21790  df-ascl 21792  df-psr 21846  df-mvr 21847  df-mpl 21848  df-opsr 21850  df-psr1 22092  df-vr1 22093  df-ply1 22094  df-coe1 22095  df-mamu 22306  df-mat 22323  df-mdet 22500  df-madu 22549  df-cpmat 22621  df-mat2pmat 22622  df-cpmat2mat 22623  df-decpmat 22678  df-pm2mp 22708  df-chpmat 22742

This theorem is referenced by: (None)