Theorem cayleyhamilton1 20697

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 20695, 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	|- .x. = ( .s ` P )
cayleyhamilton1.e	|- E = (.g ` (mulGrp ` P ) )
cayleyhamilton1.z	|- Z = ( 0g ` R )

Assertion

Ref	Expression
cayleyhamilton1	|- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) -> ( A gsumg ( n e. NN0 |-> ( ( 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   .x. , 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 20695	. . 3 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = .0. )
9	8	adantr 481	. 2 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = .0. )
10		nfv 1843	. . . . . . . 8 |- F/ n ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) )
11		nfcv 2764	. . . . . . . . . 10 |- F/_ n P
12		nfcv 2764	. . . . . . . . . 10 |- F/_ n gsumg
13		nfmpt1 4747	. . . . . . . . . 10 |- F/_ n ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) )
14	11, 12, 13	nfov 6676	. . . . . . . . 9 |- F/_ n ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) )
15	14	nfeq2 2780	. . . . . . . 8 |- F/ n ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) )
16	10, 15	nfan 1828	. . . . . . 7 |- F/ n ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) /\ ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) )
17		crngring 18558	. . . . . . . . . . . . 13 |- ( R e. CRing -> R e. Ring )
18	17	3ad2ant2 1083	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> R e. Ring )
19	18	adantr 481	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> R e. Ring )
20		cayleyhamilton1.p	. . . . . . . . . . . . 13 |- P = (Poly1 ` R )
21		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` P ) = ( Base ` P )
22	4, 1, 2, 20, 21	chpmatply1 20637	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( C ` M ) e. ( Base ` P ) )
23	22	adantr 481	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( C ` M ) e. ( 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 |- .x. = ( .s ` P )
28		eqid 2622	. . . . . . . . . . . 12 |- ( 0g ` R ) = ( 0g ` R )
29		elmapi 7879	. . . . . . . . . . . . . 14 |- ( F e. ( L ^m NN0 ) -> F : NN0 --> L )
30		ffvelrn 6357	. . . . . . . . . . . . . . 15 |- ( ( F : NN0 --> L /\ n e. NN0 ) -> ( F ` n ) e. L )
31	30	ralrimiva 2966	. . . . . . . . . . . . . 14 |- ( F : NN0 --> L -> A. n e. NN0 ( F ` n ) e. L )
32	29, 31	syl 17	. . . . . . . . . . . . 13 |- ( F e. ( L ^m NN0 ) -> A. n e. NN0 ( F ` n ) e. L )
33	32	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> A. n e. NN0 ( F ` n ) e. L )
34	29	feqmptd 6249	. . . . . . . . . . . . . . 15 |- ( F e. ( L ^m NN0 ) -> F = ( n e. NN0 |-> ( F ` n ) ) )
35		cayleyhamilton1.z	. . . . . . . . . . . . . . . 16 |- Z = ( 0g ` R )
36	35	a1i 11	. . . . . . . . . . . . . . 15 |- ( F e. ( L ^m NN0 ) -> Z = ( 0g ` R ) )
37	34, 36	breq12d 4666	. . . . . . . . . . . . . 14 |- ( F e. ( L ^m NN0 ) -> ( F finSupp Z <-> ( n e. NN0 |-> ( F ` n ) ) finSupp ( 0g ` R ) ) )
38	37	biimpa 501	. . . . . . . . . . . . 13 |- ( ( F e. ( L ^m NN0 ) /\ F finSupp Z ) -> ( n e. NN0 |-> ( F ` n ) ) finSupp ( 0g ` R ) )
39	38	adantl 482	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( n e. NN0 |-> ( F ` n ) ) finSupp ( 0g ` R ) )
40	20, 21, 24, 25, 19, 26, 27, 28, 33, 39	gsumsmonply1 19673	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) e. ( Base ` P ) )
41		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( i = n -> ( F ` i ) = ( F ` n ) )
42		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( i = n -> ( i E X ) = ( n E X ) )
43	41, 42	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( i = n -> ( ( F ` i ) .x. ( i E X ) ) = ( ( F ` n ) .x. ( n E X ) ) )
44	43	cbvmptv 4750	. . . . . . . . . . . . . 14 |- ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) = ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) )
45	44	oveq2i 6661	. . . . . . . . . . . . 13 |- ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) )
46	45	fveq2i 6194	. . . . . . . . . . . 12 |- (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) = (coe1 ` ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) )
47	20, 21, 5, 46	ply1coe1eq 19668	. . . . . . . . . . 11 |- ( ( R e. Ring /\ ( C ` M ) e. ( Base ` P ) /\ ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) e. ( Base ` P ) ) -> ( A. m e. NN0 ( K ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) <-> ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) ) )
48	19, 23, 40, 47	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( A. m e. NN0 ( K ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) <-> ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) ) )
49		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( m = n -> ( K ` m ) = ( K ` n ) )
50		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( m = n -> ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) )
51	49, 50	eqeq12d 2637	. . . . . . . . . . . . . 14 |- ( m = n -> ( ( K ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) <-> ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) ) )
52	51	rspcva 3307	. . . . . . . . . . . . 13 |- ( ( n e. NN0 /\ A. m e. NN0 ( K ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) ) -> ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) )
53		simpl 473	. . . . . . . . . . . . . . . . 17 |- ( ( ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) /\ ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) ) -> ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) )
54	18	ad2antrl 764	. . . . . . . . . . . . . . . . . . . 20 |- ( ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) -> R e. Ring )
55		ffvelrn 6357	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F : NN0 --> L /\ i e. NN0 ) -> ( F ` i ) e. L )
56	55	ralrimiva 2966	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( F : NN0 --> L -> A. i e. NN0 ( F ` i ) e. L )
57	29, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( F e. ( L ^m NN0 ) -> A. i e. NN0 ( F ` i ) e. L )
58	57	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> A. i e. NN0 ( F ` i ) e. L )
59	58	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) -> A. i e. NN0 ( F ` i ) e. L )
60	29	feqmptd 6249	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F e. ( L ^m NN0 ) -> F = ( i e. NN0 |-> ( F ` i ) ) )
61	60	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( F e. ( L ^m NN0 ) -> ( F finSupp Z <-> ( i e. NN0 |-> ( F ` i ) ) finSupp Z ) )
62	61	biimpa 501	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( F e. ( L ^m NN0 ) /\ F finSupp Z ) -> ( i e. NN0 |-> ( F ` i ) ) finSupp Z )
63	62	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( i e. NN0 |-> ( F ` i ) ) finSupp Z )
64	63	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) -> ( i e. NN0 |-> ( F ` i ) ) finSupp Z )
65		simpl 473	. . . . . . . . . . . . . . . . . . . 20 |- ( ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) -> n e. NN0 )
66	20, 21, 24, 25, 54, 26, 27, 35, 59, 64, 65	gsummoncoe1 19674	. . . . . . . . . . . . . . . . . . 19 |- ( ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) -> ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) = [_ n / i ]_ ( F ` i ) )
67		csbfv 6233	. . . . . . . . . . . . . . . . . . 19 |- [_ n / i ]_ ( F ` i ) = ( F ` n )
68	66, 67	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 |- ( ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) -> ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) = ( F ` n ) )
69	68	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) /\ ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) ) -> ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) = ( F ` n ) )
70	53, 69	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( ( ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) /\ ( n e. NN0 /\ ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) ) ) -> ( K ` n ) = ( F ` n ) )
71	70	exp32 631	. . . . . . . . . . . . . . 15 |- ( ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) -> ( n e. NN0 -> ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( K ` n ) = ( F ` n ) ) ) )
72	71	com12 32	. . . . . . . . . . . . . 14 |- ( n e. NN0 -> ( ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) -> ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( K ` n ) = ( F ` n ) ) ) )
73	72	adantr 481	. . . . . . . . . . . . 13 |- ( ( n e. NN0 /\ A. m e. NN0 ( K ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) ) -> ( ( K ` n ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` n ) -> ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( K ` n ) = ( F ` n ) ) ) )
74	52, 73	mpd 15	. . . . . . . . . . . 12 |- ( ( n e. NN0 /\ A. m e. NN0 ( K ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) ) -> ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( K ` n ) = ( F ` n ) ) )
75	74	com12 32	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( ( n e. NN0 /\ A. m e. NN0 ( K ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) ) -> ( K ` n ) = ( F ` n ) ) )
76	75	expcomd 454	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( A. m e. NN0 ( K ` m ) = ( (coe1 ` ( P gsumg ( i e. NN0 |-> ( ( F ` i ) .x. ( i E X ) ) ) ) ) ` m ) -> ( n e. NN0 -> ( K ` n ) = ( F ` n ) ) ) )
77	48, 76	sylbird 250	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) -> ( n e. NN0 -> ( K ` n ) = ( F ` n ) ) ) )
78	77	imp31 448	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) /\ ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) ) /\ n e. NN0 ) -> ( K ` n ) = ( F ` n ) )
79	78	oveq1d 6665	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) /\ ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) ) /\ n e. NN0 ) -> ( ( K ` n ) .* ( n .^ M ) ) = ( ( F ` n ) .* ( n .^ M ) ) )
80	16, 79	mpteq2da 4743	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) /\ ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) ) -> ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) = ( n e. NN0 |-> ( ( F ` n ) .* ( n .^ M ) ) ) )
81	80	oveq2d 6666	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) /\ ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) ) -> ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = ( A gsumg ( n e. NN0 |-> ( ( F ` n ) .* ( n .^ M ) ) ) ) )
82	81	eqeq1d 2624	. . . 4 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) /\ ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) ) -> ( ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = .0. <-> ( A gsumg ( n e. NN0 |-> ( ( F ` n ) .* ( n .^ M ) ) ) ) = .0. ) )
83	82	biimpd 219	. . 3 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) /\ ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) ) -> ( ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = .0. -> ( A gsumg ( n e. NN0 |-> ( ( F ` n ) .* ( n .^ M ) ) ) ) = .0. ) )
84	83	ex 450	. 2 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) -> ( ( A gsumg ( n e. NN0 |-> ( ( K ` n ) .* ( n .^ M ) ) ) ) = .0. -> ( A gsumg ( n e. NN0 |-> ( ( F ` n ) .* ( n .^ M ) ) ) ) = .0. ) ) )
85	9, 84	mpid 44	1 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( F e. ( L ^m NN0 ) /\ F finSupp Z ) ) -> ( ( C ` M ) = ( P gsumg ( n e. NN0 |-> ( ( F ` n ) .x. ( n E X ) ) ) ) -> ( A gsumg ( n e. NN0 |-> ( ( F ` n ) .* ( n .^ M ) ) ) ) = .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   [_csb 3533   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   NN0cn0 11292   Basecbs 15857   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   CharPlyMat cchpmat 20631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-cur 7393  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-symg 17798  df-pmtr 17862  df-psgn 17911  df-evpm 17912  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-assa 19312  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-mdet 20391  df-madu 20440  df-cpmat 20511  df-mat2pmat 20512  df-cpmat2mat 20513  df-decpmat 20568  df-pm2mp 20598  df-chpmat 20632

This theorem is referenced by: (None)