Theorem chpmat1d 20641

Description: The characteristic polynomial of a matrix with dimension 1. (Contributed by AV, 7-Aug-2019.)

Hypotheses

Ref	Expression
chpmat1d.c	|- C = ( N CharPlyMat R )
chpmat1d.p	|- P = (Poly1 ` R )
chpmat1d.a	|- A = ( N Mat R )
chpmat1d.b	|- B = ( Base ` A )
chpmat1d.x	|- X = (var1 ` R )
chpmat1d.z	|- .- = ( -g ` P )
chpmat1d.s	|- S = (algSc ` P )

Assertion

Ref	Expression
chpmat1d	|- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( C ` M ) = ( X .- ( S ` ( I M I ) ) ) )

Proof of Theorem chpmat1d

Step	Hyp	Ref	Expression
1		snfi 8038	. . . . . 6 |- { I } e. Fin
2		eleq1 2689	. . . . . 6 |- ( N = { I } -> ( N e. Fin <-> { I } e. Fin ) )
3	1, 2	mpbiri 248	. . . . 5 |- ( N = { I } -> N e. Fin )
4	3	adantr 481	. . . 4 |- ( ( N = { I } /\ I e. V ) -> N e. Fin )
5	4	3ad2ant2 1083	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> N e. Fin )
6		simp1 1061	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> R e. CRing )
7		simp3 1063	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> M e. B )
8		chpmat1d.c	. . . 4 |- C = ( N CharPlyMat R )
9		chpmat1d.a	. . . 4 |- A = ( N Mat R )
10		chpmat1d.b	. . . 4 |- B = ( Base ` A )
11		chpmat1d.p	. . . 4 |- P = (Poly1 ` R )
12		eqid 2622	. . . 4 |- ( N Mat P ) = ( N Mat P )
13		eqid 2622	. . . 4 |- ( N maDet P ) = ( N maDet P )
14		eqid 2622	. . . 4 |- ( -g ` ( N Mat P ) ) = ( -g ` ( N Mat P ) )
15		eqid 2622	. . . 4 |- (var1 ` R ) = (var1 ` R )
16		eqid 2622	. . . 4 |- ( .s ` ( N Mat P ) ) = ( .s ` ( N Mat P ) )
17		eqid 2622	. . . 4 |- ( N matToPolyMat R ) = ( N matToPolyMat R )
18		eqid 2622	. . . 4 |- ( 1r ` ( N Mat P ) ) = ( 1r ` ( N Mat P ) )
19	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	chpmatval 20636	. . 3 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( C ` M ) = ( ( N maDet P ) ` ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) ) )
20	5, 6, 7, 19	syl3anc 1326	. 2 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( C ` M ) = ( ( N maDet P ) ` ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) ) )
21	11	ply1crng 19568	. . . . 5 |- ( R e. CRing -> P e. CRing )
22	21	3ad2ant1 1082	. . . 4 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> P e. CRing )
23		simp2 1062	. . . 4 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( N = { I } /\ I e. V ) )
24		crngring 18558	. . . . . . . . 9 |- ( R e. CRing -> R e. Ring )
25	11	ply1ring 19618	. . . . . . . . 9 |- ( R e. Ring -> P e. Ring )
26	24, 25	syl 17	. . . . . . . 8 |- ( R e. CRing -> P e. Ring )
27	26	3ad2ant1 1082	. . . . . . 7 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> P e. Ring )
28	12	matring 20249	. . . . . . 7 |- ( ( N e. Fin /\ P e. Ring ) -> ( N Mat P ) e. Ring )
29	5, 27, 28	syl2anc 693	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( N Mat P ) e. Ring )
30		ringgrp 18552	. . . . . 6 |- ( ( N Mat P ) e. Ring -> ( N Mat P ) e. Grp )
31	29, 30	syl 17	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( N Mat P ) e. Grp )
32	12	matlmod 20235	. . . . . . 7 |- ( ( N e. Fin /\ P e. Ring ) -> ( N Mat P ) e. LMod )
33	5, 27, 32	syl2anc 693	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( N Mat P ) e. LMod )
34	24	3ad2ant1 1082	. . . . . . . 8 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> R e. Ring )
35		eqid 2622	. . . . . . . . 9 |- (Poly1 ` R ) = (Poly1 ` R )
36		eqid 2622	. . . . . . . . 9 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (Poly1 ` R ) )
37	15, 35, 36	vr1cl 19587	. . . . . . . 8 |- ( R e. Ring -> (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) )
38	34, 37	syl 17	. . . . . . 7 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) )
39	35	ply1crng 19568	. . . . . . . . . . 11 |- ( R e. CRing -> (Poly1 ` R ) e. CRing )
40	39	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (Poly1 ` R ) e. CRing )
41	11	oveq2i 6661	. . . . . . . . . . 11 |- ( N Mat P ) = ( N Mat (Poly1 ` R ) )
42	41	matsca2 20226	. . . . . . . . . 10 |- ( ( N e. Fin /\ (Poly1 ` R ) e. CRing ) -> (Poly1 ` R ) = (Scalar ` ( N Mat P ) ) )
43	5, 40, 42	syl2anc 693	. . . . . . . . 9 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (Poly1 ` R ) = (Scalar ` ( N Mat P ) ) )
44	43	eqcomd 2628	. . . . . . . 8 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (Scalar ` ( N Mat P ) ) = (Poly1 ` R ) )
45	44	fveq2d 6195	. . . . . . 7 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( Base ` (Scalar ` ( N Mat P ) ) ) = ( Base ` (Poly1 ` R ) ) )
46	38, 45	eleqtrrd 2704	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (var1 ` R ) e. ( Base ` (Scalar ` ( N Mat P ) ) ) )
47		eqid 2622	. . . . . . . 8 |- ( Base ` ( N Mat P ) ) = ( Base ` ( N Mat P ) )
48	47, 18	ringidcl 18568	. . . . . . 7 |- ( ( N Mat P ) e. Ring -> ( 1r ` ( N Mat P ) ) e. ( Base ` ( N Mat P ) ) )
49	29, 48	syl 17	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( 1r ` ( N Mat P ) ) e. ( Base ` ( N Mat P ) ) )
50		eqid 2622	. . . . . . 7 |- (Scalar ` ( N Mat P ) ) = (Scalar ` ( N Mat P ) )
51		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` ( N Mat P ) ) ) = ( Base ` (Scalar ` ( N Mat P ) ) )
52	47, 50, 16, 51	lmodvscl 18880	. . . . . 6 |- ( ( ( N Mat P ) e. LMod /\ (var1 ` R ) e. ( Base ` (Scalar ` ( N Mat P ) ) ) /\ ( 1r ` ( N Mat P ) ) e. ( Base ` ( N Mat P ) ) ) -> ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) e. ( Base ` ( N Mat P ) ) )
53	33, 46, 49, 52	syl3anc 1326	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) e. ( Base ` ( N Mat P ) ) )
54	17, 9, 10, 11, 12	mat2pmatbas 20531	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( ( N matToPolyMat R ) ` M ) e. ( Base ` ( N Mat P ) ) )
55	5, 34, 7, 54	syl3anc 1326	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( N matToPolyMat R ) ` M ) e. ( Base ` ( N Mat P ) ) )
56	47, 14	grpsubcl 17495	. . . . 5 |- ( ( ( N Mat P ) e. Grp /\ ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) e. ( Base ` ( N Mat P ) ) /\ ( ( N matToPolyMat R ) ` M ) e. ( Base ` ( N Mat P ) ) ) -> ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) e. ( Base ` ( N Mat P ) ) )
57	31, 53, 55, 56	syl3anc 1326	. . . 4 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) e. ( Base ` ( N Mat P ) ) )
58	13, 12, 47	m1detdiag 20403	. . . 4 |- ( ( P e. CRing /\ ( N = { I } /\ I e. V ) /\ ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) e. ( Base ` ( N Mat P ) ) ) -> ( ( N maDet P ) ` ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) ) = ( I ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) I ) )
59	22, 23, 57, 58	syl3anc 1326	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( N maDet P ) ` ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) ) = ( I ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) I ) )
60		chpmat1d.x	. . . . . . . . 9 |- X = (var1 ` R )
61	60	eqcomi 2631	. . . . . . . 8 |- (var1 ` R ) = X
62	61	a1i 11	. . . . . . 7 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> (var1 ` R ) = X )
63	62	oveq1d 6665	. . . . . 6 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) = ( X ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) )
64	63	oveq1d 6665	. . . . 5 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) = ( ( X ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) )
65	64	oveqd 6667	. . . 4 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) I ) = ( I ( ( X ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) I ) )
66		chpmat1d.z	. . . . . 6 |- .- = ( -g ` P )
67		chpmat1d.s	. . . . . 6 |- S = (algSc ` P )
68	8, 11, 9, 10, 60, 66, 67, 12, 17	chpmat1dlem 20640	. . . . 5 |- ( ( R e. Ring /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I ( ( X ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) I ) = ( X .- ( S ` ( I M I ) ) ) )
69	24, 68	syl3an1 1359	. . . 4 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I ( ( X ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) I ) = ( X .- ( S ` ( I M I ) ) ) )
70	65, 69	eqtrd 2656	. . 3 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( I ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) I ) = ( X .- ( S ` ( I M I ) ) ) )
71	59, 70	eqtrd 2656	. 2 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( ( N maDet P ) ` ( ( (var1 ` R ) ( .s ` ( N Mat P ) ) ( 1r ` ( N Mat P ) ) ) ( -g ` ( N Mat P ) ) ( ( N matToPolyMat R ) ` M ) ) ) = ( X .- ( S ` ( I M I ) ) ) )
72	20, 71	eqtrd 2656	1 |- ( ( R e. CRing /\ ( N = { I } /\ I e. V ) /\ M e. B ) -> ( C ` M ) = ( X .- ( S ` ( I M I ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   Grpcgrp 17422   -gcsg 17424   1rcur 18501   Ringcrg 18547   CRingccrg 18548   LModclmod 18863  algSccascl 19311  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-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-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-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-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  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-cnfld 19747  df-zring 19819  df-zrh 19852  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-mdet 20391  df-mat2pmat 20512  df-chpmat 20632

This theorem is referenced by: (None)