Definition df-edring 35553

Description: Define division ring on trace-preserving endomorphisms. The multiplication operation is reversed composition, per the definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)

Assertion

Ref	Expression
df-edring	|- EDRing = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) >. } ) )

Distinct variable group:   w, k, f, s, t

Detailed syntax breakdown of Definition df-edring

Step	Hyp	Ref	Expression
1		cedring 35549	. 2 class EDRing
2		vk	. . 3 setvar k
3		cvv 3113	. . 3 class _V
4		vw	. . . 4 setvar w
5	2	cv 1378	. . . . 5 class k
6		clh 34780	. . . . 5 class LHyp
7	5, 6	cfv 5586	. . . 4 class ( LHyp ` k )
8		cnx 14480	. . . . . . 7 class ndx
9		cbs 14483	. . . . . . 7 class Base
10	8, 9	cfv 5586	. . . . . 6 class ( Base ` ndx )
11	4	cv 1378	. . . . . . 7 class w
12		ctendo 35548	. . . . . . . 8 class TEndo
13	5, 12	cfv 5586	. . . . . . 7 class ( TEndo ` k )
14	11, 13	cfv 5586	. . . . . 6 class ( ( TEndo ` k ) ` w )
15	10, 14	cop 4033	. . . . 5 class <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >.
16		cplusg 14548	. . . . . . 7 class +g
17	8, 16	cfv 5586	. . . . . 6 class ( +g ` ndx )
18		vs	. . . . . . 7 setvar s
19		vt	. . . . . . 7 setvar t
20		vf	. . . . . . . 8 setvar f
21		cltrn 34897	. . . . . . . . . 10 class LTrn
22	5, 21	cfv 5586	. . . . . . . . 9 class ( LTrn ` k )
23	11, 22	cfv 5586	. . . . . . . 8 class ( ( LTrn ` k ) ` w )
24	20	cv 1378	. . . . . . . . . 10 class f
25	18	cv 1378	. . . . . . . . . 10 class s
26	24, 25	cfv 5586	. . . . . . . . 9 class ( s ` f )
27	19	cv 1378	. . . . . . . . . 10 class t
28	24, 27	cfv 5586	. . . . . . . . 9 class ( t ` f )
29	26, 28	ccom 5003	. . . . . . . 8 class ( ( s ` f ) o. ( t ` f ) )
30	20, 23, 29	cmpt 4505	. . . . . . 7 class ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) )
31	18, 19, 14, 14, 30	cmpt2 6284	. . . . . 6 class ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) )
32	17, 31	cop 4033	. . . . 5 class <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >.
33		cmulr 14549	. . . . . . 7 class .r
34	8, 33	cfv 5586	. . . . . 6 class ( .r ` ndx )
35	25, 27	ccom 5003	. . . . . . 7 class ( s o. t )
36	18, 19, 14, 14, 35	cmpt2 6284	. . . . . 6 class ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) )
37	34, 36	cop 4033	. . . . 5 class <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) >.
38	15, 32, 37	ctp 4031	. . . 4 class { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) >. }
39	4, 7, 38	cmpt 4505	. . 3 class ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) >. } )
40	2, 3, 39	cmpt 4505	. 2 class ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) >. } ) )
41	1, 40	wceq 1379	1 wff EDRing = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) >. } ) )

This definition is referenced by:  erngfset  35595