Theorem trlfset 35447

Description: The set of all traces of lattice translations for a lattice K. (Contributed by NM, 20-May-2012.)

Hypotheses

Ref	Expression
trlset.b	|- B = ( Base ` K )
trlset.l	|- .<_ = ( le ` K )
trlset.j	|- .\/ = ( join ` K )
trlset.m	|- ./\ = ( meet ` K )
trlset.a	|- A = ( Atoms ` K )
trlset.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
trlfset	|- ( K e. C -> ( trL ` K ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) )

Distinct variable groups:   A, p   x, B   w, H   f, p, w, x, K

Allowed substitution hints:   A( x, w, f)   B( w, f, p)   C( x, w, f, p)   H( x, f, p)   .\/ ( x, w, f, p)   .<_ ( x, w, f, p)   ./\ ( x, w, f, p)

Proof of Theorem trlfset

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. C -> K e. _V )
2		fveq2 6191	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		trlset.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . 6 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6	5	fveq1d 6193	. . . . 5 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7		fveq2 6191	. . . . . . 7 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
8		trlset.b	. . . . . . 7 |- B = ( Base ` K )
9	7, 8	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( Base ` k ) = B )
10		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
11		trlset.a	. . . . . . . 8 |- A = ( Atoms ` K )
12	10, 11	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( Atoms ` k ) = A )
13		fveq2 6191	. . . . . . . . . . 11 |- ( k = K -> ( le ` k ) = ( le ` K ) )
14		trlset.l	. . . . . . . . . . 11 |- .<_ = ( le ` K )
15	13, 14	syl6eqr 2674	. . . . . . . . . 10 |- ( k = K -> ( le ` k ) = .<_ )
16	15	breqd 4664	. . . . . . . . 9 |- ( k = K -> ( p ( le ` k ) w <-> p .<_ w ) )
17	16	notbid 308	. . . . . . . 8 |- ( k = K -> ( -. p ( le ` k ) w <-> -. p .<_ w ) )
18		fveq2 6191	. . . . . . . . . . 11 |- ( k = K -> ( meet ` k ) = ( meet ` K ) )
19		trlset.m	. . . . . . . . . . 11 |- ./\ = ( meet ` K )
20	18, 19	syl6eqr 2674	. . . . . . . . . 10 |- ( k = K -> ( meet ` k ) = ./\ )
21		fveq2 6191	. . . . . . . . . . . 12 |- ( k = K -> ( join ` k ) = ( join ` K ) )
22		trlset.j	. . . . . . . . . . . 12 |- .\/ = ( join ` K )
23	21, 22	syl6eqr 2674	. . . . . . . . . . 11 |- ( k = K -> ( join ` k ) = .\/ )
24	23	oveqd 6667	. . . . . . . . . 10 |- ( k = K -> ( p ( join ` k ) ( f ` p ) ) = ( p .\/ ( f ` p ) ) )
25		eqidd 2623	. . . . . . . . . 10 |- ( k = K -> w = w )
26	20, 24, 25	oveq123d 6671	. . . . . . . . 9 |- ( k = K -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( p .\/ ( f ` p ) ) ./\ w ) )
27	26	eqeq2d 2632	. . . . . . . 8 |- ( k = K -> ( x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) <-> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) )
28	17, 27	imbi12d 334	. . . . . . 7 |- ( k = K -> ( ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) <-> ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) )
29	12, 28	raleqbidv 3152	. . . . . 6 |- ( k = K -> ( A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) <-> A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) )
30	9, 29	riotaeqbidv 6614	. . . . 5 |- ( k = K -> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) = ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) )
31	6, 30	mpteq12dv 4733	. . . 4 |- ( k = K -> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) = ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) )
32	4, 31	mpteq12dv 4733	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) )
33		df-trl 35446	. . 3 |- trL = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) ) )
34		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
35	3, 34	eqeltri 2697	. . . 4 |- H e. _V
36	35	mptex 6486	. . 3 |- ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) e. _V
37	32, 33, 36	fvmpt 6282	. 2 |- ( K e. _V -> ( trL ` K ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) )
38	1, 37	syl 17	1 |- ( K e. C -> ( trL ` K ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-trl 35446

This theorem is referenced by:  trlset  35448