Theorem chmatval 20634

Description: The entries of the characteristic matrix of a matrix. (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 10-Dec-2019.)

Hypotheses

Ref	Expression
chmatcl.a	|- A = ( N Mat R )
chmatcl.b	|- B = ( Base ` A )
chmatcl.p	|- P = (Poly1 ` R )
chmatcl.y	|- Y = ( N Mat P )
chmatcl.x	|- X = (var1 ` R )
chmatcl.t	|- T = ( N matToPolyMat R )
chmatcl.s	|- .- = ( -g ` Y )
chmatcl.m	|- .x. = ( .s ` Y )
chmatcl.1	|- .1. = ( 1r ` Y )
chmatcl.h	|- H = ( ( X .x. .1. ) .- ( T ` M ) )
chmatval.s	|- .~ = ( -g ` P )
chmatval.0	|- .0. = ( 0g ` P )

Assertion

Ref	Expression
chmatval	|- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I H J ) = if ( I = J , ( X .~ ( I ( T ` M ) J ) ) , ( .0. .~ ( I ( T ` M ) J ) ) ) )

Proof of Theorem chmatval

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		chmatcl.h	. . . 4 |- H = ( ( X .x. .1. ) .- ( T ` M ) )
2	1	oveqi 6663	. . 3 |- ( I H J ) = ( I ( ( X .x. .1. ) .- ( T ` M ) ) J )
3		chmatcl.p	. . . . . . 7 |- P = (Poly1 ` R )
4	3	ply1ring 19618	. . . . . 6 |- ( R e. Ring -> P e. Ring )
5	4	3ad2ant2 1083	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> P e. Ring )
6	5	adantr 481	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> P e. Ring )
7	4	anim2i 593	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( N e. Fin /\ P e. Ring ) )
8	7	3adant3 1081	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( N e. Fin /\ P e. Ring ) )
9		chmatcl.x	. . . . . . . 8 |- X = (var1 ` R )
10		eqid 2622	. . . . . . . 8 |- ( Base ` P ) = ( Base ` P )
11	9, 3, 10	vr1cl 19587	. . . . . . 7 |- ( R e. Ring -> X e. ( Base ` P ) )
12	11	3ad2ant2 1083	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> X e. ( Base ` P ) )
13		chmatcl.y	. . . . . . . . 9 |- Y = ( N Mat P )
14	3, 13	pmatring 20498	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> Y e. Ring )
15	14	3adant3 1081	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> Y e. Ring )
16		eqid 2622	. . . . . . . 8 |- ( Base ` Y ) = ( Base ` Y )
17		chmatcl.1	. . . . . . . 8 |- .1. = ( 1r ` Y )
18	16, 17	ringidcl 18568	. . . . . . 7 |- ( Y e. Ring -> .1. e. ( Base ` Y ) )
19	15, 18	syl 17	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> .1. e. ( Base ` Y ) )
20		chmatcl.m	. . . . . . 7 |- .x. = ( .s ` Y )
21	10, 13, 16, 20	matvscl 20237	. . . . . 6 |- ( ( ( N e. Fin /\ P e. Ring ) /\ ( X e. ( Base ` P ) /\ .1. e. ( Base ` Y ) ) ) -> ( X .x. .1. ) e. ( Base ` Y ) )
22	8, 12, 19, 21	syl12anc 1324	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( X .x. .1. ) e. ( Base ` Y ) )
23	22	adantr 481	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( X .x. .1. ) e. ( Base ` Y ) )
24		chmatcl.t	. . . . . 6 |- T = ( N matToPolyMat R )
25		chmatcl.a	. . . . . 6 |- A = ( N Mat R )
26		chmatcl.b	. . . . . 6 |- B = ( Base ` A )
27	24, 25, 26, 3, 13	mat2pmatbas 20531	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( T ` M ) e. ( Base ` Y ) )
28	27	adantr 481	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( T ` M ) e. ( Base ` Y ) )
29		simpr 477	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I e. N /\ J e. N ) )
30		chmatcl.s	. . . . 5 |- .- = ( -g ` Y )
31		chmatval.s	. . . . 5 |- .~ = ( -g ` P )
32	13, 16, 30, 31	matsubgcell 20240	. . . 4 |- ( ( P e. Ring /\ ( ( X .x. .1. ) e. ( Base ` Y ) /\ ( T ` M ) e. ( Base ` Y ) ) /\ ( I e. N /\ J e. N ) ) -> ( I ( ( X .x. .1. ) .- ( T ` M ) ) J ) = ( ( I ( X .x. .1. ) J ) .~ ( I ( T ` M ) J ) ) )
33	6, 23, 28, 29, 32	syl121anc 1331	. . 3 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I ( ( X .x. .1. ) .- ( T ` M ) ) J ) = ( ( I ( X .x. .1. ) J ) .~ ( I ( T ` M ) J ) ) )
34	2, 33	syl5eq 2668	. 2 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I H J ) = ( ( I ( X .x. .1. ) J ) .~ ( I ( T ` M ) J ) ) )
35	17	a1i 11	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> .1. = ( 1r ` Y ) )
36	35	oveq2d 6666	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( X .x. .1. ) = ( X .x. ( 1r ` Y ) ) )
37		simpl 473	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> N e. Fin )
38	4	adantl 482	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> P e. Ring )
39	11	adantl 482	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> X e. ( Base ` P ) )
40	37, 38, 39	3jca 1242	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> ( N e. Fin /\ P e. Ring /\ X e. ( Base ` P ) ) )
41	40	3adant3 1081	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( N e. Fin /\ P e. Ring /\ X e. ( Base ` P ) ) )
42	41	adantr 481	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( N e. Fin /\ P e. Ring /\ X e. ( Base ` P ) ) )
43		chmatval.0	. . . . . . . 8 |- .0. = ( 0g ` P )
44	13, 10, 20, 43	matsc 20256	. . . . . . 7 |- ( ( N e. Fin /\ P e. Ring /\ X e. ( Base ` P ) ) -> ( X .x. ( 1r ` Y ) ) = ( i e. N , j e. N |-> if ( i = j , X , .0. ) ) )
45	42, 44	syl 17	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( X .x. ( 1r ` Y ) ) = ( i e. N , j e. N |-> if ( i = j , X , .0. ) ) )
46	36, 45	eqtrd 2656	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( X .x. .1. ) = ( i e. N , j e. N |-> if ( i = j , X , .0. ) ) )
47		eqeq12 2635	. . . . . . 7 |- ( ( i = I /\ j = J ) -> ( i = j <-> I = J ) )
48	47	ifbid 4108	. . . . . 6 |- ( ( i = I /\ j = J ) -> if ( i = j , X , .0. ) = if ( I = J , X , .0. ) )
49	48	adantl 482	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) /\ ( i = I /\ j = J ) ) -> if ( i = j , X , .0. ) = if ( I = J , X , .0. ) )
50		simprl 794	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> I e. N )
51		simpr 477	. . . . . 6 |- ( ( I e. N /\ J e. N ) -> J e. N )
52	51	adantl 482	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> J e. N )
53		fvex 6201	. . . . . . . 8 |- (var1 ` R ) e. _V
54	9, 53	eqeltri 2697	. . . . . . 7 |- X e. _V
55		fvex 6201	. . . . . . . 8 |- ( 0g ` P ) e. _V
56	43, 55	eqeltri 2697	. . . . . . 7 |- .0. e. _V
57	54, 56	ifex 4156	. . . . . 6 |- if ( I = J , X , .0. ) e. _V
58	57	a1i 11	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> if ( I = J , X , .0. ) e. _V )
59	46, 49, 50, 52, 58	ovmpt2d 6788	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I ( X .x. .1. ) J ) = if ( I = J , X , .0. ) )
60	59	oveq1d 6665	. . 3 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( ( I ( X .x. .1. ) J ) .~ ( I ( T ` M ) J ) ) = ( if ( I = J , X , .0. ) .~ ( I ( T ` M ) J ) ) )
61		ovif 6737	. . 3 |- ( if ( I = J , X , .0. ) .~ ( I ( T ` M ) J ) ) = if ( I = J , ( X .~ ( I ( T ` M ) J ) ) , ( .0. .~ ( I ( T ` M ) J ) ) )
62	60, 61	syl6eq 2672	. 2 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( ( I ( X .x. .1. ) J ) .~ ( I ( T ` M ) J ) ) = if ( I = J , ( X .~ ( I ( T ` M ) J ) ) , ( .0. .~ ( I ( T ` M ) J ) ) ) )
63	34, 62	eqtrd 2656	1 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I H J ) = if ( I = J , ( X .~ ( I ( T ` M ) J ) ) , ( .0. .~ ( I ( T ` M ) J ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   Basecbs 15857   .scvsca 15945   0gc0g 16100   -gcsg 17424   1rcur 18501   Ringcrg 18547  var₁cv1 19546  Poly₁cpl1 19547   Mat cmat 20213   matToPolyMat cmat2pmat 20509

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  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-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-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-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  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-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  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-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  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-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-mat2pmat 20512

This theorem is referenced by: (None)