Theorem chpmatfval 20635

Description: Value of the characteristic polynomial function. (Contributed by AV, 2-Aug-2019.)

Hypotheses

Ref	Expression
chpmatfval.c	|- C = ( N CharPlyMat R )
chpmatfval.a	|- A = ( N Mat R )
chpmatfval.b	|- B = ( Base ` A )
chpmatfval.p	|- P = (Poly1 ` R )
chpmatfval.y	|- Y = ( N Mat P )
chpmatfval.d	|- D = ( N maDet P )
chpmatfval.s	|- .- = ( -g ` Y )
chpmatfval.x	|- X = (var1 ` R )
chpmatfval.m	|- .x. = ( .s ` Y )
chpmatfval.t	|- T = ( N matToPolyMat R )
chpmatfval.i	|- .1. = ( 1r ` Y )

Assertion

Ref	Expression
chpmatfval	|- ( ( N e. Fin /\ R e. V ) -> C = ( m e. B |-> ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) ) )

Distinct variable groups:   B, m   D, m   .1. , m   m, N   R, m   m, X   T, m   .x. , m   .- , m

Allowed substitution hints:   A( m)   C( m)   P( m)   V( m)   Y( m)

Proof of Theorem chpmatfval

Dummy variables n r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		chpmatfval.c	. 2 |- C = ( N CharPlyMat R )
2		df-chpmat 20632	. . . 4 |- CharPlyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( ( n maDet (Poly1 ` r ) ) ` ( ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) ( -g ` ( n Mat (Poly1 ` r ) ) ) ( ( n matToPolyMat r ) ` m ) ) ) ) )
3	2	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. V ) -> CharPlyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( ( n maDet (Poly1 ` r ) ) ` ( ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) ( -g ` ( n Mat (Poly1 ` r ) ) ) ( ( n matToPolyMat r ) ` m ) ) ) ) ) )
4		oveq12 6659	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
5		chpmatfval.a	. . . . . . . 8 |- A = ( N Mat R )
6	4, 5	syl6eqr 2674	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = A )
7	6	fveq2d 6195	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` A ) )
8		chpmatfval.b	. . . . . 6 |- B = ( Base ` A )
9	7, 8	syl6eqr 2674	. . . . 5 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
10		simpl 473	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> n = N )
11		simpr 477	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> r = R )
12	11	fveq2d 6195	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> (Poly1 ` r ) = (Poly1 ` R ) )
13		chpmatfval.p	. . . . . . . . 9 |- P = (Poly1 ` R )
14	12, 13	syl6eqr 2674	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> (Poly1 ` r ) = P )
15	10, 14	oveq12d 6668	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n maDet (Poly1 ` r ) ) = ( N maDet P ) )
16		chpmatfval.d	. . . . . . 7 |- D = ( N maDet P )
17	15, 16	syl6eqr 2674	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( n maDet (Poly1 ` r ) ) = D )
18		fveq2 6191	. . . . . . . . . . . . 13 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
19	18	adantl 482	. . . . . . . . . . . 12 |- ( ( n = N /\ r = R ) -> (Poly1 ` r ) = (Poly1 ` R ) )
20	19, 13	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( n = N /\ r = R ) -> (Poly1 ` r ) = P )
21	10, 20	oveq12d 6668	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> ( n Mat (Poly1 ` r ) ) = ( N Mat P ) )
22		chpmatfval.y	. . . . . . . . . 10 |- Y = ( N Mat P )
23	21, 22	syl6eqr 2674	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( n Mat (Poly1 ` r ) ) = Y )
24	23	fveq2d 6195	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( -g ` ( n Mat (Poly1 ` r ) ) ) = ( -g ` Y ) )
25		chpmatfval.s	. . . . . . . 8 |- .- = ( -g ` Y )
26	24, 25	syl6eqr 2674	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( -g ` ( n Mat (Poly1 ` r ) ) ) = .- )
27	23	fveq2d 6195	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( .s ` ( n Mat (Poly1 ` r ) ) ) = ( .s ` Y ) )
28		chpmatfval.m	. . . . . . . . 9 |- .x. = ( .s ` Y )
29	27, 28	syl6eqr 2674	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( .s ` ( n Mat (Poly1 ` r ) ) ) = .x. )
30		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> (var1 ` r ) = (var1 ` R ) )
31		chpmatfval.x	. . . . . . . . . 10 |- X = (var1 ` R )
32	30, 31	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> (var1 ` r ) = X )
33	32	adantl 482	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> (var1 ` r ) = X )
34	23	fveq2d 6195	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( 1r ` ( n Mat (Poly1 ` r ) ) ) = ( 1r ` Y ) )
35		chpmatfval.i	. . . . . . . . 9 |- .1. = ( 1r ` Y )
36	34, 35	syl6eqr 2674	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( 1r ` ( n Mat (Poly1 ` r ) ) ) = .1. )
37	29, 33, 36	oveq123d 6671	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) = ( X .x. .1. ) )
38		oveq12 6659	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( n matToPolyMat r ) = ( N matToPolyMat R ) )
39		chpmatfval.t	. . . . . . . . 9 |- T = ( N matToPolyMat R )
40	38, 39	syl6eqr 2674	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( n matToPolyMat r ) = T )
41	40	fveq1d 6193	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( ( n matToPolyMat r ) ` m ) = ( T ` m ) )
42	26, 37, 41	oveq123d 6671	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) ( -g ` ( n Mat (Poly1 ` r ) ) ) ( ( n matToPolyMat r ) ` m ) ) = ( ( X .x. .1. ) .- ( T ` m ) ) )
43	17, 42	fveq12d 6197	. . . . 5 |- ( ( n = N /\ r = R ) -> ( ( n maDet (Poly1 ` r ) ) ` ( ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) ( -g ` ( n Mat (Poly1 ` r ) ) ) ( ( n matToPolyMat r ) ` m ) ) ) = ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) )
44	9, 43	mpteq12dv 4733	. . . 4 |- ( ( n = N /\ r = R ) -> ( m e. ( Base ` ( n Mat r ) ) |-> ( ( n maDet (Poly1 ` r ) ) ` ( ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) ( -g ` ( n Mat (Poly1 ` r ) ) ) ( ( n matToPolyMat r ) ` m ) ) ) ) = ( m e. B |-> ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) ) )
45	44	adantl 482	. . 3 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> ( m e. ( Base ` ( n Mat r ) ) |-> ( ( n maDet (Poly1 ` r ) ) ` ( ( (var1 ` r ) ( .s ` ( n Mat (Poly1 ` r ) ) ) ( 1r ` ( n Mat (Poly1 ` r ) ) ) ) ( -g ` ( n Mat (Poly1 ` r ) ) ) ( ( n matToPolyMat r ) ` m ) ) ) ) = ( m e. B |-> ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) ) )
46		simpl 473	. . 3 |- ( ( N e. Fin /\ R e. V ) -> N e. Fin )
47		elex 3212	. . . 4 |- ( R e. V -> R e. _V )
48	47	adantl 482	. . 3 |- ( ( N e. Fin /\ R e. V ) -> R e. _V )
49		fvex 6201	. . . . 5 |- ( Base ` A ) e. _V
50	8, 49	eqeltri 2697	. . . 4 |- B e. _V
51		mptexg 6484	. . . 4 |- ( B e. _V -> ( m e. B |-> ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) ) e. _V )
52	50, 51	mp1i 13	. . 3 |- ( ( N e. Fin /\ R e. V ) -> ( m e. B |-> ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) ) e. _V )
53	3, 45, 46, 48, 52	ovmpt2d 6788	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( N CharPlyMat R ) = ( m e. B |-> ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) ) )
54	1, 53	syl5eq 2668	1 |- ( ( N e. Fin /\ R e. V ) -> C = ( m e. B |-> ( D ` ( ( X .x. .1. ) .- ( T ` m ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   Basecbs 15857   .scvsca 15945   -gcsg 17424   1rcur 18501  var₁cv1 19546  Poly₁cpl1 19547   Mat cmat 20213   maDet cmdat 20390   matToPolyMat cmat2pmat 20509   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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-chpmat 20632

This theorem is referenced by:  chpmatval  20636