Definition df-trnN 35393

Description: Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)

Assertion

Ref	Expression
df-trnN	|- Trn = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) } ) )

Distinct variable group:   k, d, f, q, r

Detailed syntax breakdown of Definition df-trnN

Step	Hyp	Ref	Expression
1		ctrnN 35389	. 2 class Trn
2		vk	. . 3 setvar k
3		cvv 3200	. . 3 class _V
4		vd	. . . 4 setvar d
5	2	cv 1482	. . . . 5 class k
6		catm 34550	. . . . 5 class Atoms
7	5, 6	cfv 5888	. . . 4 class ( Atoms ` k )
8		vq	. . . . . . . . . . 11 setvar q
9	8	cv 1482	. . . . . . . . . 10 class q
10		vf	. . . . . . . . . . . 12 setvar f
11	10	cv 1482	. . . . . . . . . . 11 class f
12	9, 11	cfv 5888	. . . . . . . . . 10 class ( f ` q )
13		cpadd 35081	. . . . . . . . . . 11 class +P
14	5, 13	cfv 5888	. . . . . . . . . 10 class ( +P ` k )
15	9, 12, 14	co 6650	. . . . . . . . 9 class ( q ( +P ` k ) ( f ` q ) )
16	4	cv 1482	. . . . . . . . . . 11 class d
17	16	csn 4177	. . . . . . . . . 10 class { d }
18		cpolN 35188	. . . . . . . . . . 11 class _|_P
19	5, 18	cfv 5888	. . . . . . . . . 10 class ( _|_P ` k )
20	17, 19	cfv 5888	. . . . . . . . 9 class ( ( _|_P ` k ) ` { d } )
21	15, 20	cin 3573	. . . . . . . 8 class ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) )
22		vr	. . . . . . . . . . 11 setvar r
23	22	cv 1482	. . . . . . . . . 10 class r
24	23, 11	cfv 5888	. . . . . . . . . 10 class ( f ` r )
25	23, 24, 14	co 6650	. . . . . . . . 9 class ( r ( +P ` k ) ( f ` r ) )
26	25, 20	cin 3573	. . . . . . . 8 class ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) )
27	21, 26	wceq 1483	. . . . . . 7 wff ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) )
28		cwpointsN 35272	. . . . . . . . 9 class WAtoms
29	5, 28	cfv 5888	. . . . . . . 8 class ( WAtoms ` k )
30	16, 29	cfv 5888	. . . . . . 7 class ( ( WAtoms ` k ) ` d )
31	27, 22, 30	wral 2912	. . . . . 6 wff A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) )
32	31, 8, 30	wral 2912	. . . . 5 wff A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) )
33		cdilN 35388	. . . . . . 7 class Dil
34	5, 33	cfv 5888	. . . . . 6 class ( Dil ` k )
35	16, 34	cfv 5888	. . . . 5 class ( ( Dil ` k ) ` d )
36	32, 10, 35	crab 2916	. . . 4 class { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) }
37	4, 7, 36	cmpt 4729	. . 3 class ( d e. ( Atoms ` k ) |-> { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) } )
38	2, 3, 37	cmpt 4729	. 2 class ( k e. _V |-> ( d e. ( Atoms ` k ) |-> { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) } ) )
39	1, 38	wceq 1483	1 wff Trn = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> { f e. ( ( Dil ` k ) ` d ) | A. q e. ( ( WAtoms ` k ) ` d ) A. r e. ( ( WAtoms ` k ) ` d ) ( ( q ( +P ` k ) ( f ` q ) ) i^i ( ( _|_P ` k ) ` { d } ) ) = ( ( r ( +P ` k ) ( f ` r ) ) i^i ( ( _|_P ` k ) ` { d } ) ) } ) )

This definition is referenced by:  trnfsetN  35442