Theorem chpscmatgsummon 20650

Description: The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of scaled monomials. (Contributed by AV, 2-Sep-2019.)

Hypotheses

Ref	Expression
chp0mat.c	|- C = ( N CharPlyMat R )
chp0mat.p	|- P = (Poly1 ` R )
chp0mat.a	|- A = ( N Mat R )
chp0mat.x	|- X = (var1 ` R )
chp0mat.g	|- G = (mulGrp ` P )
chp0mat.m	|- .^ = (.g ` G )
chpscmat.d	|- D = { m e. ( Base ` A ) | E. c e. ( Base ` R ) A. i e. N A. j e. N ( i m j ) = if ( i = j , c , ( 0g ` R ) ) }
chpscmat.s	|- S = (algSc ` P )
chpscmat.m	|- .- = ( -g ` P )
chpscmatgsum.f	|- F = (.g ` P )
chpscmatgsum.h	|- H = (mulGrp ` R )
chpscmatgsum.e	|- E = (.g ` H )
chpscmatgsum.i	|- I = ( invg ` R )
chpscmatgsum.s	|- .x. = ( .s ` P )
chpscmatgsum.z	|- Z = (.g ` R )

Assertion

Ref	Expression
chpscmatgsummon	|- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( C ` M ) = ( P gsumg ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( ( # ` N ) _C l ) Z ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) ) ) )

Distinct variable groups:   i, j, A   i, N, j   P, i, j   R, i, j   i, X, j   A, c, m   D, n   n, E   n, I   M, c, i, j, m, n   N, c, m, n   P, n   R, c, m, n   S, n   D, l   F, l   I, l   J, l, n   M, l   N, l   P, l   R, l   S, l   X, l   .^ , l

Allowed substitution hints:   A( n, l)   C( i, j, m, n, c, l)   D( i, j, m, c)   P( m, c)   S( i, j, m, c)   .x. ( i, j, m, n, c, l)   E( i, j, m, c, l)   .^ ( i, j, m, n, c)   F( i, j, m, n, c)   G( i, j, m, n, c, l)   H( i, j, m, n, c, l)   I( i, j, m, c)   J( i, j, m, c)   .- ( i, j, m, n, c, l)   X( m, n, c)   Z( i, j, m, n, c, l)

Proof of Theorem chpscmatgsummon

Step	Hyp	Ref	Expression
1		chp0mat.c	. . 3 |- C = ( N CharPlyMat R )
2		chp0mat.p	. . 3 |- P = (Poly1 ` R )
3		chp0mat.a	. . 3 |- A = ( N Mat R )
4		chp0mat.x	. . 3 |- X = (var1 ` R )
5		chp0mat.g	. . 3 |- G = (mulGrp ` P )
6		chp0mat.m	. . 3 |- .^ = (.g ` G )
7		chpscmat.d	. . 3 |- D = { m e. ( Base ` A ) | E. c e. ( Base ` R ) A. i e. N A. j e. N ( i m j ) = if ( i = j , c , ( 0g ` R ) ) }
8		chpscmat.s	. . 3 |- S = (algSc ` P )
9		chpscmat.m	. . 3 |- .- = ( -g ` P )
10		chpscmatgsum.f	. . 3 |- F = (.g ` P )
11		chpscmatgsum.h	. . 3 |- H = (mulGrp ` R )
12		chpscmatgsum.e	. . 3 |- E = (.g ` H )
13		chpscmatgsum.i	. . 3 |- I = ( invg ` R )
14		chpscmatgsum.s	. . 3 |- .x. = ( .s ` P )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	chpscmatgsumbin 20649	. 2 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( C ` M ) = ( P gsumg ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( # ` N ) _C l ) F ( ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) .x. ( l .^ X ) ) ) ) ) )
16		crngring 18558	. . . . . . . . 9 |- ( R e. CRing -> R e. Ring )
17	16	adantl 482	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing ) -> R e. Ring )
18	2	ply1lmod 19622	. . . . . . . 8 |- ( R e. Ring -> P e. LMod )
19	17, 18	syl 17	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing ) -> P e. LMod )
20	19	ad2antrr 762	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> P e. LMod )
21	11	ringmgp 18553	. . . . . . . . . 10 |- ( R e. Ring -> H e. Mnd )
22	17, 21	syl 17	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing ) -> H e. Mnd )
23	22	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> H e. Mnd )
24		fznn0sub 12373	. . . . . . . . 9 |- ( l e. ( 0 ... ( # ` N ) ) -> ( ( # ` N ) - l ) e. NN0 )
25	24	adantl 482	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( ( # ` N ) - l ) e. NN0 )
26		ringgrp 18552	. . . . . . . . . . . . 13 |- ( R e. Ring -> R e. Grp )
27	16, 26	syl 17	. . . . . . . . . . . 12 |- ( R e. CRing -> R e. Grp )
28	27	adantl 482	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. CRing ) -> R e. Grp )
29	28	adantr 481	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> R e. Grp )
30		simp2 1062	. . . . . . . . . . . . 13 |- ( ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) -> J e. N )
31		elrabi 3359	. . . . . . . . . . . . . . 15 |- ( M e. { m e. ( Base ` A ) | E. c e. ( Base ` R ) A. i e. N A. j e. N ( i m j ) = if ( i = j , c , ( 0g ` R ) ) } -> M e. ( Base ` A ) )
32	31, 7	eleq2s 2719	. . . . . . . . . . . . . 14 |- ( M e. D -> M e. ( Base ` A ) )
33	32	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) -> M e. ( Base ` A ) )
34	30, 30, 33	3jca 1242	. . . . . . . . . . . 12 |- ( ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) -> ( J e. N /\ J e. N /\ M e. ( Base ` A ) ) )
35	34	adantl 482	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( J e. N /\ J e. N /\ M e. ( Base ` A ) ) )
36		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
37	3, 36	matecl 20231	. . . . . . . . . . 11 |- ( ( J e. N /\ J e. N /\ M e. ( Base ` A ) ) -> ( J M J ) e. ( Base ` R ) )
38	35, 37	syl 17	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( J M J ) e. ( Base ` R ) )
39	36, 13	grpinvcl 17467	. . . . . . . . . 10 |- ( ( R e. Grp /\ ( J M J ) e. ( Base ` R ) ) -> ( I ` ( J M J ) ) e. ( Base ` R ) )
40	29, 38, 39	syl2anc 693	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( I ` ( J M J ) ) e. ( Base ` R ) )
41	40	adantr 481	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( I ` ( J M J ) ) e. ( Base ` R ) )
42	11, 36	mgpbas 18495	. . . . . . . . 9 |- ( Base ` R ) = ( Base ` H )
43	42, 12	mulgnn0cl 17558	. . . . . . . 8 |- ( ( H e. Mnd /\ ( ( # ` N ) - l ) e. NN0 /\ ( I ` ( J M J ) ) e. ( Base ` R ) ) -> ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) e. ( Base ` R ) )
44	23, 25, 41, 43	syl3anc 1326	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) e. ( Base ` R ) )
45	2	ply1sca 19623	. . . . . . . . . . 11 |- ( R e. CRing -> R = (Scalar ` P ) )
46	45	adantl 482	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing ) -> R = (Scalar ` P ) )
47	46	eqcomd 2628	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing ) -> (Scalar ` P ) = R )
48	47	fveq2d 6195	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing ) -> ( Base ` (Scalar ` P ) ) = ( Base ` R ) )
49	48	ad2antrr 762	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( Base ` (Scalar ` P ) ) = ( Base ` R ) )
50	44, 49	eleqtrrd 2704	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) e. ( Base ` (Scalar ` P ) ) )
51		hashcl 13147	. . . . . . . 8 |- ( N e. Fin -> ( # ` N ) e. NN0 )
52	51	ad2antrr 762	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( # ` N ) e. NN0 )
53		elfzelz 12342	. . . . . . 7 |- ( l e. ( 0 ... ( # ` N ) ) -> l e. ZZ )
54		bccl 13109	. . . . . . 7 |- ( ( ( # ` N ) e. NN0 /\ l e. ZZ ) -> ( ( # ` N ) _C l ) e. NN0 )
55	52, 53, 54	syl2an 494	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( ( # ` N ) _C l ) e. NN0 )
56	2	ply1ring 19618	. . . . . . . . . 10 |- ( R e. Ring -> P e. Ring )
57	5	ringmgp 18553	. . . . . . . . . 10 |- ( P e. Ring -> G e. Mnd )
58	16, 56, 57	3syl 18	. . . . . . . . 9 |- ( R e. CRing -> G e. Mnd )
59	58	adantl 482	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing ) -> G e. Mnd )
60	59	ad2antrr 762	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> G e. Mnd )
61		elfznn0 12433	. . . . . . . 8 |- ( l e. ( 0 ... ( # ` N ) ) -> l e. NN0 )
62	61	adantl 482	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> l e. NN0 )
63		eqid 2622	. . . . . . . . . 10 |- ( Base ` P ) = ( Base ` P )
64	4, 2, 63	vr1cl 19587	. . . . . . . . 9 |- ( R e. Ring -> X e. ( Base ` P ) )
65	17, 64	syl 17	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing ) -> X e. ( Base ` P ) )
66	65	ad2antrr 762	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> X e. ( Base ` P ) )
67	5, 63	mgpbas 18495	. . . . . . . 8 |- ( Base ` P ) = ( Base ` G )
68	67, 6	mulgnn0cl 17558	. . . . . . 7 |- ( ( G e. Mnd /\ l e. NN0 /\ X e. ( Base ` P ) ) -> ( l .^ X ) e. ( Base ` P ) )
69	60, 62, 66, 68	syl3anc 1326	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( l .^ X ) e. ( Base ` P ) )
70		eqid 2622	. . . . . . 7 |- (Scalar ` P ) = (Scalar ` P )
71		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
72		eqid 2622	. . . . . . 7 |- (.g ` (Scalar ` P ) ) = (.g ` (Scalar ` P ) )
73	63, 70, 14, 71, 10, 72	lmodvsmmulgdi 18898	. . . . . 6 |- ( ( P e. LMod /\ ( ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) e. ( Base ` (Scalar ` P ) ) /\ ( ( # ` N ) _C l ) e. NN0 /\ ( l .^ X ) e. ( Base ` P ) ) ) -> ( ( ( # ` N ) _C l ) F ( ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) .x. ( l .^ X ) ) ) = ( ( ( ( # ` N ) _C l ) (.g ` (Scalar ` P ) ) ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) )
74	20, 50, 55, 69, 73	syl13anc 1328	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( ( ( # ` N ) _C l ) F ( ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) .x. ( l .^ X ) ) ) = ( ( ( ( # ` N ) _C l ) (.g ` (Scalar ` P ) ) ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) )
75		chpscmatgsum.z	. . . . . . . . 9 |- Z = (.g ` R )
76	46	fveq2d 6195	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing ) -> (.g ` R ) = (.g ` (Scalar ` P ) ) )
77	75, 76	syl5req 2669	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing ) -> (.g ` (Scalar ` P ) ) = Z )
78	77	ad2antrr 762	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> (.g ` (Scalar ` P ) ) = Z )
79	78	oveqd 6667	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( ( ( # ` N ) _C l ) (.g ` (Scalar ` P ) ) ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) = ( ( ( # ` N ) _C l ) Z ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) )
80	79	oveq1d 6665	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( ( ( ( # ` N ) _C l ) (.g ` (Scalar ` P ) ) ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) = ( ( ( ( # ` N ) _C l ) Z ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) )
81	74, 80	eqtrd 2656	. . . 4 |- ( ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) /\ l e. ( 0 ... ( # ` N ) ) ) -> ( ( ( # ` N ) _C l ) F ( ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) .x. ( l .^ X ) ) ) = ( ( ( ( # ` N ) _C l ) Z ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) )
82	81	mpteq2dva 4744	. . 3 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( # ` N ) _C l ) F ( ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) .x. ( l .^ X ) ) ) ) = ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( ( # ` N ) _C l ) Z ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) ) )
83	82	oveq2d 6666	. 2 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( P gsumg ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( # ` N ) _C l ) F ( ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) .x. ( l .^ X ) ) ) ) ) = ( P gsumg ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( ( # ` N ) _C l ) Z ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) ) ) )
84	15, 83	eqtrd 2656	1 |- ( ( ( N e. Fin /\ R e. CRing ) /\ ( M e. D /\ J e. N /\ A. n e. N ( n M n ) = ( J M J ) ) ) -> ( C ` M ) = ( P gsumg ( l e. ( 0 ... ( # ` N ) ) |-> ( ( ( ( # ` N ) _C l ) Z ( ( ( # ` N ) - l ) E ( I ` ( J M J ) ) ) ) .x. ( l .^ X ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   - cmin 10266   NN0cn0 11292   ZZcz 11377   ...cfz 12326   _C cbc 13089   #chash 13117   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548   LModclmod 18863  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547   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-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-fac 13061  df-bc 13090  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-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-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)