Theorem chpmat0d 20639

Description: The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)

Hypothesis

Ref	Expression
chpmat0.c	|- C = ( (/) CharPlyMat R )

Assertion

Ref	Expression
chpmat0d	|- ( R e. Ring -> ( C ` (/) ) = ( 1r ` (Poly1 ` R ) ) )

Proof of Theorem chpmat0d

Step	Hyp	Ref	Expression
1		0fin 8188	. . . 4 |- (/) e. Fin
2	1	a1i 11	. . 3 |- ( R e. Ring -> (/) e. Fin )
3		id 22	. . 3 |- ( R e. Ring -> R e. Ring )
4		0ex 4790	. . . . 5 |- (/) e. _V
5	4	snid 4208	. . . 4 |- (/) e. { (/) }
6		mat0dimbas0 20272	. . . 4 |- ( R e. Ring -> ( Base ` ( (/) Mat R ) ) = { (/) } )
7	5, 6	syl5eleqr 2708	. . 3 |- ( R e. Ring -> (/) e. ( Base ` ( (/) Mat R ) ) )
8		chpmat0.c	. . . 4 |- C = ( (/) CharPlyMat R )
9		eqid 2622	. . . 4 |- ( (/) Mat R ) = ( (/) Mat R )
10		eqid 2622	. . . 4 |- ( Base ` ( (/) Mat R ) ) = ( Base ` ( (/) Mat R ) )
11		eqid 2622	. . . 4 |- (Poly1 ` R ) = (Poly1 ` R )
12		eqid 2622	. . . 4 |- ( (/) Mat (Poly1 ` R ) ) = ( (/) Mat (Poly1 ` R ) )
13		eqid 2622	. . . 4 |- ( (/) maDet (Poly1 ` R ) ) = ( (/) maDet (Poly1 ` R ) )
14		eqid 2622	. . . 4 |- ( -g ` ( (/) Mat (Poly1 ` R ) ) ) = ( -g ` ( (/) Mat (Poly1 ` R ) ) )
15		eqid 2622	. . . 4 |- (var1 ` R ) = (var1 ` R )
16		eqid 2622	. . . 4 |- ( .s ` ( (/) Mat (Poly1 ` R ) ) ) = ( .s ` ( (/) Mat (Poly1 ` R ) ) )
17		eqid 2622	. . . 4 |- ( (/) matToPolyMat R ) = ( (/) matToPolyMat R )
18		eqid 2622	. . . 4 |- ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) = ( 1r ` ( (/) Mat (Poly1 ` R ) ) )
19	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	chpmatval 20636	. . 3 |- ( ( (/) e. Fin /\ R e. Ring /\ (/) e. ( Base ` ( (/) Mat R ) ) ) -> ( C ` (/) ) = ( ( (/) maDet (Poly1 ` R ) ) ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) )
20	2, 3, 7, 19	syl3anc 1326	. 2 |- ( R e. Ring -> ( C ` (/) ) = ( ( (/) maDet (Poly1 ` R ) ) ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) )
21	11	ply1ring 19618	. . . 4 |- ( R e. Ring -> (Poly1 ` R ) e. Ring )
22		mdet0pr 20398	. . . . 5 |- ( (Poly1 ` R ) e. Ring -> ( (/) maDet (Poly1 ` R ) ) = { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } )
23	22	fveq1d 6193	. . . 4 |- ( (Poly1 ` R ) e. Ring -> ( ( (/) maDet (Poly1 ` R ) ) ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) = ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) )
24	21, 23	syl 17	. . 3 |- ( R e. Ring -> ( ( (/) maDet (Poly1 ` R ) ) ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) = ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) )
25	12	mat0dimid 20274	. . . . . . . . . 10 |- ( (Poly1 ` R ) e. Ring -> ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) = (/) )
26	21, 25	syl 17	. . . . . . . . 9 |- ( R e. Ring -> ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) = (/) )
27	26	oveq2d 6666	. . . . . . . 8 |- ( R e. Ring -> ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) = ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) (/) ) )
28		eqid 2622	. . . . . . . . . 10 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (Poly1 ` R ) )
29	15, 11, 28	vr1cl 19587	. . . . . . . . 9 |- ( R e. Ring -> (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) )
30	12	mat0dimscm 20275	. . . . . . . . 9 |- ( ( (Poly1 ` R ) e. Ring /\ (var1 ` R ) e. ( Base ` (Poly1 ` R ) ) ) -> ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) (/) ) = (/) )
31	21, 29, 30	syl2anc 693	. . . . . . . 8 |- ( R e. Ring -> ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) (/) ) = (/) )
32	27, 31	eqtrd 2656	. . . . . . 7 |- ( R e. Ring -> ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) = (/) )
33		d0mat2pmat 20543	. . . . . . 7 |- ( R e. Ring -> ( ( (/) matToPolyMat R ) ` (/) ) = (/) )
34	32, 33	oveq12d 6668	. . . . . 6 |- ( R e. Ring -> ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) = ( (/) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) (/) ) )
35	12	matring 20249	. . . . . . . . 9 |- ( ( (/) e. Fin /\ (Poly1 ` R ) e. Ring ) -> ( (/) Mat (Poly1 ` R ) ) e. Ring )
36	1, 21, 35	sylancr 695	. . . . . . . 8 |- ( R e. Ring -> ( (/) Mat (Poly1 ` R ) ) e. Ring )
37		ringgrp 18552	. . . . . . . 8 |- ( ( (/) Mat (Poly1 ` R ) ) e. Ring -> ( (/) Mat (Poly1 ` R ) ) e. Grp )
38	36, 37	syl 17	. . . . . . 7 |- ( R e. Ring -> ( (/) Mat (Poly1 ` R ) ) e. Grp )
39		mat0dimbas0 20272	. . . . . . . . 9 |- ( (Poly1 ` R ) e. Ring -> ( Base ` ( (/) Mat (Poly1 ` R ) ) ) = { (/) } )
40	21, 39	syl 17	. . . . . . . 8 |- ( R e. Ring -> ( Base ` ( (/) Mat (Poly1 ` R ) ) ) = { (/) } )
41	5, 40	syl5eleqr 2708	. . . . . . 7 |- ( R e. Ring -> (/) e. ( Base ` ( (/) Mat (Poly1 ` R ) ) ) )
42		eqid 2622	. . . . . . . 8 |- ( Base ` ( (/) Mat (Poly1 ` R ) ) ) = ( Base ` ( (/) Mat (Poly1 ` R ) ) )
43		eqid 2622	. . . . . . . 8 |- ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) = ( 0g ` ( (/) Mat (Poly1 ` R ) ) )
44	42, 43, 14	grpsubid 17499	. . . . . . 7 |- ( ( ( (/) Mat (Poly1 ` R ) ) e. Grp /\ (/) e. ( Base ` ( (/) Mat (Poly1 ` R ) ) ) ) -> ( (/) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) (/) ) = ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) )
45	38, 41, 44	syl2anc 693	. . . . . 6 |- ( R e. Ring -> ( (/) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) (/) ) = ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) )
46	34, 45	eqtrd 2656	. . . . 5 |- ( R e. Ring -> ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) = ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) )
47	46	fveq2d 6195	. . . 4 |- ( R e. Ring -> ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) = ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) ) )
48	12	mat0dim0 20273	. . . . . . 7 |- ( (Poly1 ` R ) e. Ring -> ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) = (/) )
49	21, 48	syl 17	. . . . . 6 |- ( R e. Ring -> ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) = (/) )
50	49	fveq2d 6195	. . . . 5 |- ( R e. Ring -> ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) ) = ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` (/) ) )
51		fvex 6201	. . . . . 6 |- ( 1r ` (Poly1 ` R ) ) e. _V
52	4, 51	fvsn 6446	. . . . 5 |- ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` (/) ) = ( 1r ` (Poly1 ` R ) )
53	50, 52	syl6eq 2672	. . . 4 |- ( R e. Ring -> ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` ( 0g ` ( (/) Mat (Poly1 ` R ) ) ) ) = ( 1r ` (Poly1 ` R ) ) )
54	47, 53	eqtrd 2656	. . 3 |- ( R e. Ring -> ( { <. (/) , ( 1r ` (Poly1 ` R ) ) >. } ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) = ( 1r ` (Poly1 ` R ) ) )
55	24, 54	eqtrd 2656	. 2 |- ( R e. Ring -> ( ( (/) maDet (Poly1 ` R ) ) ` ( ( (var1 ` R ) ( .s ` ( (/) Mat (Poly1 ` R ) ) ) ( 1r ` ( (/) Mat (Poly1 ` R ) ) ) ) ( -g ` ( (/) Mat (Poly1 ` R ) ) ) ( ( (/) matToPolyMat R ) ` (/) ) ) ) = ( 1r ` (Poly1 ` R ) ) )
56	20, 55	eqtrd 2656	1 |- ( R e. Ring -> ( C ` (/) ) = ( 1r ` (Poly1 ` R ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   (/)c0 3915   {csn 4177   <.cop 4183   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   1rcur 18501   Ringcrg 18547  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-evpm 17912  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  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-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)