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 479	. 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 1909	. . . . . . . 8 ⊢ Ⅎ𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))
11		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑛𝑃
12		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑛 Σg
13		nfmpt1 5252	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
14	11, 12, 13	nfov 7443	. . . . . . . . 9 ⊢ Ⅎ𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
15	14	nfeq2 2910	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
16	10, 15	nfan 1894	. . . . . . 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 20184	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18	17	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
19	18	adantr 479	. . . . . . . . . . 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 2725	. . . . . . . . . . . . 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 479	. . . . . . . . . . 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 2725	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
29		elmapi 8861	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹:ℕ0⟶𝐿)
30		ffvelcdm 7084	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑛 ∈ ℕ0) → (𝐹‘𝑛) ∈ 𝐿)
31	30	ralrimiva 3136	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑛 ∈ ℕ0 (𝐹‘𝑛) ∈ 𝐿)
34	29	feqmptd 6960	. . . . . . . . . . . . . . 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 5157	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → (𝐹 finSupp 𝑍 ↔ (𝑛 ∈ ℕ0 ↦ (𝐹‘𝑛)) finSupp (0g‘𝑅)))
38	37	biimpa 475	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍) → (𝑛 ∈ ℕ0 ↦ (𝐹‘𝑛)) finSupp (0g‘𝑅))
39	38	adantl 480	. . . . . . . . . . . 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 22230	. . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
42		oveq1 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
43	41, 42	oveq12d 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → ((𝐹‘𝑖) · (𝑖𝐸𝑋)) = ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
44	43	cbvmptv 5257	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))
45	44	oveq2i 7424	. . . . . . . . . . . . 13 ⊢ (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋))))
46	45	fveq2i 6893	. . . . . . . . . . . 12 ⊢ (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))))
47	20, 21, 5, 46	ply1coe1eq 22223	. . . . . . . . . . 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 1368	. . . . . . . . . 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 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝐾‘𝑚) = (𝐾‘𝑛))
50		fveq2 6890	. . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . 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 3601	. . . . . . . . . . . . 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 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)))) → (𝐾‘𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹‘𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
54	18	ad2antrl 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → 𝑅 ∈ Ring)
55		ffvelcdm 7084	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:ℕ0⟶𝐿 ∧ 𝑖 ∈ ℕ0) → (𝐹‘𝑖) ∈ 𝐿)
56	55	ralrimiva 3136	. . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
59	58	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → ∀𝑖 ∈ ℕ0 (𝐹‘𝑖) ∈ 𝐿)
60	29	feqmptd 6960	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)))
61	60	breq1d 5154	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (𝐿 ↑m ℕ0) → (𝐹 finSupp 𝑍 ↔ (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍))
62	61	biimpa 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
63	62	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
64	63	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍))) → (𝑖 ∈ ℕ0 ↦ (𝐹‘𝑖)) finSupp 𝑍)
65		simpl 481	. . . . . . . . . . . . . . . . . . . 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 22231	. . . . . . . . . . . . . . . . . . 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 6940	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋𝑛 / 𝑖⦌(𝐹‘𝑖) = (𝐹‘𝑛)
68	66, 67	eqtrdi 2781	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 415	. . . . . . . . . 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 259	. . . . . . . . 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 416	. . . . . . . 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 7428	. . . . . . 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 5242	. . . . . 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 7429	. . . . 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 2727	. . . 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 228	. . 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 411	. 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 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ⦋csb 3886   class class class wbr 5144   ↦ cmpt 5227  ⟶wf 6539  ‘cfv 6543  (class class class)co 7413   ↑_m cmap 8838  Fincfn 8957   finSupp cfsupp 9380  ℕ₀cn0 12497  Basecbs 17174   ·_𝑠 cvsca 17231  0_gc0g 17415   Σ_g cgsu 17416  ._gcmg 19022  mulGrpcmgp 20073  Ringcrg 20172  CRingccrg 20173  var₁cv1 22098  Poly₁cpl1 22099  coe₁cco1 22100   Mat cmat 22320   CharPlyMat cchpmat 22741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-addf 11212  ax-mulf 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-ofr 7680  df-om 7866  df-1st 7987  df-2nd 7988  df-supp 8159  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 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9381  df-sup 9460  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-xnn0 12570  df-z 12584  df-dec 12703  df-uz 12848  df-rp 13002  df-fz 13512  df-fzo 13655  df-seq 13994  df-exp 14054  df-hash 14317  df-word 14492  df-lsw 14540  df-concat 14548  df-s1 14573  df-substr 14618  df-pfx 14648  df-splice 14727  df-reverse 14736  df-s2 14826  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-0g 17417  df-gsum 17418  df-prds 17423  df-pws 17425  df-mre 17560  df-mrc 17561  df-acs 17563  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-mhm 18734  df-submnd 18735  df-efmnd 18820  df-grp 18892  df-minusg 18893  df-sbg 18894  df-mulg 19023  df-subg 19077  df-ghm 19167  df-gim 19212  df-cntz 19267  df-oppg 19296  df-symg 19321  df-pmtr 19396  df-psgn 19445  df-evpm 19446  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-srg 20126  df-ring 20174  df-cring 20175  df-oppr 20272  df-dvdsr 20295  df-unit 20296  df-invr 20326  df-dvr 20339  df-rhm 20410  df-subrng 20482  df-subrg 20507  df-drng 20625  df-lmod 20744  df-lss 20815  df-sra 21057  df-rgmod 21058  df-cnfld 21279  df-zring 21372  df-zrh 21428  df-dsmm 21665  df-frlm 21680  df-assa 21786  df-ascl 21788  df-psr 21841  df-mvr 21842  df-mpl 21843  df-opsr 21845  df-psr1 22102  df-vr1 22103  df-ply1 22104  df-coe1 22105  df-mamu 22304  df-mat 22321  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)