Theorem cayleyhamilton1 22812

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 22810, 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 22810	. . 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 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑃
12		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑛 Σg
13		nfmpt1 5201	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
14	11, 12, 13	nfov 7399	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
15	14	nfeq2 2909	. . . . . . . 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 20165	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘𝑃)
22	4, 1, 2, 20, 21	chpmatply1 22752	. . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
29		elmapi 8799	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹:ℕ0⟶𝐿)
30		ffvelcdm 7035	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑛 ∈ ℕ0) → (𝐹‘𝑛) ∈ 𝐿)
31	30	ralrimiva 3125	. . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . . 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 22227	. . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
42		oveq1 7376	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
43	41, 42	oveq12d 7387	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → ((𝐹‘𝑖) · (𝑖𝐸𝑋)) = ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
44	43	cbvmptv 5206	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
45	44	oveq2i 7380	. . . . . . . . . . . . 13 ⊢ (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
46	45	fveq2i 6843	. . . . . . . . . . . 12 ⊢ (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))))
47	20, 21, 5, 46	ply1coe1eq 22220	. . . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝐾‘𝑚) = (𝐾‘𝑛))
50		fveq2 6840	. . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . 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 3583	. . . . . . . . . . . . 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 7035	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑖 ∈ ℕ0) → (𝐹‘𝑖) ∈ 𝐿)
56	55	ralrimiva 3125	. . . . . . . . . . . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)))
61	60	breq1d 5112	. . . . . . . . . . . . . . . . . . . . . . 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 22228	. . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋𝑛 / 𝑖⦌(𝐹‘𝑖) = (𝐹‘𝑛)
68	66, 67	eqtrdi 2780	. . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . 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 7384	. . . . . . 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 5194	. . . . . 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 7385	. . . . 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 2731	. . . 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 2109  ∀wral 3044  ⦋csb 3859   class class class wbr 5102   ↦ cmpt 5183  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ↑_m cmap 8776  Fincfn 8895   finSupp cfsupp 9288  ℕ₀cn0 12418  Basecbs 17155   ·_𝑠 cvsca 17200  0_gc0g 17378   Σ_g cgsu 17379  ._gcmg 18981  mulGrpcmgp 20060  Ringcrg 20153  CRingccrg 20154  var₁cv1 22093  Poly₁cpl1 22094  coe₁cco1 22095   Mat cmat 22327   CharPlyMat cchpmat 22746

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-addf 11123  ax-mulf 11124

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-ofr 7634  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-tpos 8182  df-cur 8223  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-sup 9369  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-rp 12928  df-fz 13445  df-fzo 13592  df-seq 13943  df-exp 14003  df-hash 14272  df-word 14455  df-lsw 14504  df-concat 14512  df-s1 14537  df-substr 14582  df-pfx 14612  df-splice 14691  df-reverse 14700  df-s2 14790  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-0g 17380  df-gsum 17381  df-prds 17386  df-pws 17388  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-mhm 18692  df-submnd 18693  df-efmnd 18778  df-grp 18850  df-minusg 18851  df-sbg 18852  df-mulg 18982  df-subg 19037  df-ghm 19127  df-gim 19173  df-cntz 19231  df-oppg 19260  df-symg 19284  df-pmtr 19356  df-psgn 19405  df-evpm 19406  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-srg 20107  df-ring 20155  df-cring 20156  df-oppr 20257  df-dvdsr 20277  df-unit 20278  df-invr 20308  df-dvr 20321  df-rhm 20392  df-subrng 20466  df-subrg 20490  df-drng 20651  df-lmod 20800  df-lss 20870  df-sra 21112  df-rgmod 21113  df-cnfld 21297  df-zring 21389  df-zrh 21445  df-dsmm 21674  df-frlm 21689  df-assa 21795  df-ascl 21797  df-psr 21851  df-mvr 21852  df-mpl 21853  df-opsr 21855  df-psr1 22097  df-vr1 22098  df-ply1 22099  df-coe1 22100  df-mamu 22311  df-mat 22328  df-mdet 22505  df-madu 22554  df-cpmat 22626  df-mat2pmat 22627  df-cpmat2mat 22628  df-decpmat 22683  df-pm2mp 22713  df-chpmat 22747

This theorem is referenced by: (None)