Definition df-tgrp 36031

Description: Define the class of all translation groups. k is normally a member of HL. Each base set is the set of all lattice translations with respect to a hyperplane w, and the operation is function composition. Similar to definition of G in [Crawley] p. 116, third paragraph (which defines this for geomodular lattices). (Contributed by NM, 5-Jun-2013.)

Assertion

Ref	Expression
df-tgrp	|- TGrp = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. } ) )

Distinct variable group:   w, k, f, g

Detailed syntax breakdown of Definition df-tgrp

Step	Hyp	Ref	Expression
1		ctgrp 36030	. 2 class TGrp
2		vk	. . 3 setvar k
3		cvv 3200	. . 3 class _V
4		vw	. . . 4 setvar w
5	2	cv 1482	. . . . 5 class k
6		clh 35270	. . . . 5 class LHyp
7	5, 6	cfv 5888	. . . 4 class ( LHyp ` k )
8		cnx 15854	. . . . . . 7 class ndx
9		cbs 15857	. . . . . . 7 class Base
10	8, 9	cfv 5888	. . . . . 6 class ( Base ` ndx )
11	4	cv 1482	. . . . . . 7 class w
12		cltrn 35387	. . . . . . . 8 class LTrn
13	5, 12	cfv 5888	. . . . . . 7 class ( LTrn ` k )
14	11, 13	cfv 5888	. . . . . 6 class ( ( LTrn ` k ) ` w )
15	10, 14	cop 4183	. . . . 5 class <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >.
16		cplusg 15941	. . . . . . 7 class +g
17	8, 16	cfv 5888	. . . . . 6 class ( +g ` ndx )
18		vf	. . . . . . 7 setvar f
19		vg	. . . . . . 7 setvar g
20	18	cv 1482	. . . . . . . 8 class f
21	19	cv 1482	. . . . . . . 8 class g
22	20, 21	ccom 5118	. . . . . . 7 class ( f o. g )
23	18, 19, 14, 14, 22	cmpt2 6652	. . . . . 6 class ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) )
24	17, 23	cop 4183	. . . . 5 class <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >.
25	15, 24	cpr 4179	. . . 4 class { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. }
26	4, 7, 25	cmpt 4729	. . 3 class ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. } )
27	2, 3, 26	cmpt 4729	. 2 class ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. } ) )
28	1, 27	wceq 1483	1 wff TGrp = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. } ) )

This definition is referenced by:  tgrpfset  36032