Theorem cayleyhamilton1 23103

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 23101, 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 23101	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
9	8	adantr 486	. 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 1947	. . . . . . . 8 ⊢ Ⅎ𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))
11		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑃
12		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑛 Σg
13		nfmpt1 5212	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
14	11, 12, 13	nfov 7449	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
15	14	nfeq2 2944	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
16	10, 15	nfan 1932	. . . . . . 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 20375	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18	17	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
19	18	adantr 486	. . . . . . . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘𝑃)
22	4, 1, 2, 20, 21	chpmatply1 23043	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
23	22	adantr 486	. . . . . . . . . . 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 2765	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
29		elmapi 8852	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹:ℕ0⟶𝐿)
30		ffvelcdm 7080	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑛 ∈ ℕ0) → (𝐹‘𝑛) ∈ 𝐿)
31	30	ralrimiva 3159	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ0⟶𝐿 → ∀𝑛 ∈ ℕ0 (𝐹‘𝑛) ∈ 𝐿)
32	29, 31	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → ∀𝑛 ∈ ℕ0 (𝐹‘𝑛) ∈ 𝐿)
33	32	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑛 ∈ ℕ0 (𝐹‘𝑛) ∈ 𝐿)
34	29	feqmptd 6953	. . . . . . . . . . . . . . 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 5124	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → (𝐹 finSupp 𝑍 ↔ (𝑛 ∈ ℕ0 ↦ (𝐹‘𝑛)) finSupp (0g‘𝑅)))
38	37	biimpa 482	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍) → (𝑛 ∈ ℕ0 ↦ (𝐹‘𝑛)) finSupp (0g‘𝑅))
39	38	adantl 487	. . . . . . . . . . . 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 22521	. . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
42		oveq1 7426	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
43	41, 42	oveq12d 7437	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → ((𝐹‘𝑖) · (𝑖𝐸𝑋)) = ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
44	43	cbvmptv 5217	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
45	44	oveq2i 7430	. . . . . . . . . . . . 13 ⊢ (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
46	45	fveq2i 6888	. . . . . . . . . . . 12 ⊢ (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))))
47	20, 21, 5, 46	ply1coe1eq 22514	. . . . . . . . . . 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 1398	. . . . . . . . . 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 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝐾‘𝑚) = (𝐾‘𝑛))
50		fveq2 6885	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → 𝑅 ∈ Ring)
55		ffvelcdm 7080	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑖 ∈ ℕ0) → (𝐹‘𝑖) ∈ 𝐿)
56	55	ralrimiva 3159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:ℕ0⟶𝐿 → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
57	29, 56	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
58	57	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
59	58	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
60	29	feqmptd 6953	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)))
61	60	breq1d 5121	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → (𝐹 finSupp 𝑍 ↔ (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍))
62	61	biimpa 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
63	62	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
64	63	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
65		simpl 488	. . . . . . . . . . . . . . . . . . . 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 22522	. . . . . . . . . . . . . . . . . . 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 6932	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋𝑛 / 𝑖⦌(𝐹‘𝑖) = (𝐹‘𝑛)
68	66, 67	eqtrdi 2816	. . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 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 33	. . . . . . . . . . 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 422	. . . . . . . . . 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 263	. . . . . . . . 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 423	. . . . . . . 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 7434	. . . . . . 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 5205	. . . . . 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 7435	. . . . 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 2767	. . . 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 232	. . 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 418	. 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 45	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 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ⦋csb 3854   class class class wbr 5111   ↦ cmpt 5194  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ↑_m cmap 8830  Fincfn 8949   finSupp cfsupp 9328  ℕ₀cn0 12523  Basecbs 17295   ·_𝑠 cvsca 17340  0_gc0g 17518   Σ_g cgsu 17519  ._gcmg 19181  mulGrpcmgp 20264  Ringcrg 20363  CRingccrg 20364  var₁cv1 22390  Poly₁cpl1 22391  coe₁cco1 22392   Mat cmat 22618   CharPlyMat cchpmat 23037

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11175  ax-resscn 11176  ax-1cn 11177  ax-icn 11178  ax-addcl 11179  ax-addrcl 11180  ax-mulcl 11181  ax-mulrcl 11182  ax-mulcom 11183  ax-addass 11184  ax-mulass 11185  ax-distr 11186  ax-i2m1 11187  ax-1ne0 11188  ax-1rid 11189  ax-rnegex 11190  ax-rrecex 11191  ax-cnre 11192  ax-pre-lttri 11193  ax-pre-lttrn 11194  ax-pre-ltadd 11195  ax-pre-mulgt0 11196  ax-addf 11198  ax-mulf 11199

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-ofr 7685  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-tpos 8228  df-cur 8269  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-sup 9409  df-oi 9479  df-card 9941  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11462  df-neg 11463  df-div 11891  df-nn 12253  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-9 12329  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12732  df-uz 12883  df-rp 13037  df-fz 13556  df-fzo 13704  df-seq 14060  df-exp 14120  df-hash 14389  df-word 14573  df-lsw 14622  df-concat 14630  df-s1 14657  df-substr 14703  df-pfx 14735  df-splice 14813  df-reverse 14822  df-s2 14913  df-struct 17233  df-sets 17250  df-slot 17268  df-ndx 17280  df-base 17296  df-ress 17317  df-plusg 17349  df-mulr 17350  df-starv 17351  df-sca 17352  df-vsca 17353  df-ip 17354  df-tset 17355  df-ple 17356  df-ds 17358  df-unif 17359  df-hom 17360  df-cco 17361  df-0g 17520  df-gsum 17521  df-prds 17526  df-pws 17528  df-mre 17664  df-mrc 17665  df-acs 17667  df-mgm 18724  df-sgrp 18813  df-mnd 18829  df-mhm 18882  df-submnd 18883  df-efmnd 18969  df-grp 19051  df-minusg 19052  df-sbg 19053  df-mulg 19182  df-subg 19237  df-ghm 19332  df-gim 19377  df-cntz 19435  df-oppg 19464  df-symg 19488  df-pmtr 19560  df-psgn 19609  df-evpm 19610  df-cmn 19900  df-abl 19901  df-mgp 20265  df-rng 20279  df-ur 20312  df-srg 20317  df-ring 20365  df-cring 20366  df-oppr 20469  df-dvdsr 20489  df-unit 20490  df-invr 20520  df-dvr 20533  df-rhm 20604  df-subrng 20699  df-subrg 20723  df-drng 20883  df-lmod 21037  df-lss 21107  df-sra 21348  df-rgmod 21349  df-cnfld 21577  df-zring 21651  df-zrh 21707  df-dsmm 21936  df-frlm 21951  df-assa 22057  df-ascl 22059  df-psr 22113  df-mvr 22114  df-mpl 22115  df-opsr 22117  df-psr1 22394  df-vr1 22395  df-ply1 22396  df-coe1 22397  df-mamu 22602  df-mat 22619  df-mdet 22796  df-madu 22845  df-cpmat 22917  df-mat2pmat 22918  df-cpmat2mat 22919  df-decpmat 22974  df-pm2mp 23004  df-chpmat 23038

This theorem is used by: (None)