Theorem trlset 35448

Description: The set of traces of lattice translations for a fiducial co-atom W. (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 )
trlset.t	|- T = ( ( LTrn ` K ) ` W )
trlset.r	|- R = ( ( trL ` K ) ` W )

Assertion

Ref	Expression
trlset	|- ( ( K e. C /\ W e. H ) -> R = ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )

Distinct variable groups:   A, p   x, B   f, p, x, K   T, f   f, W, p, x

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

Proof of Theorem trlset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlset.r	. . 3 |- R = ( ( trL ` K ) ` W )
2		trlset.b	. . . . 5 |- B = ( Base ` K )
3		trlset.l	. . . . 5 |- .<_ = ( le ` K )
4		trlset.j	. . . . 5 |- .\/ = ( join ` K )
5		trlset.m	. . . . 5 |- ./\ = ( meet ` K )
6		trlset.a	. . . . 5 |- A = ( Atoms ` K )
7		trlset.h	. . . . 5 |- H = ( LHyp ` K )
8	2, 3, 4, 5, 6, 7	trlfset 35447	. . . 4 |- ( 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 ) ) ) ) ) )
9	8	fveq1d 6193	. . 3 |- ( K e. C -> ( ( trL ` 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 ) ) ) ) ) ` W ) )
10	1, 9	syl5eq 2668	. 2 |- ( K e. C -> R = ( ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ` W ) )
11		fveq2 6191	. . . . 5 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
12		breq2 4657	. . . . . . . . 9 |- ( w = W -> ( p .<_ w <-> p .<_ W ) )
13	12	notbid 308	. . . . . . . 8 |- ( w = W -> ( -. p .<_ w <-> -. p .<_ W ) )
14		oveq2 6658	. . . . . . . . 9 |- ( w = W -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( p .\/ ( f ` p ) ) ./\ W ) )
15	14	eqeq2d 2632	. . . . . . . 8 |- ( w = W -> ( x = ( ( p .\/ ( f ` p ) ) ./\ w ) <-> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) )
16	13, 15	imbi12d 334	. . . . . . 7 |- ( w = W -> ( ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) <-> ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
17	16	ralbidv 2986	. . . . . 6 |- ( w = W -> ( A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) <-> A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
18	17	riotabidv 6613	. . . . 5 |- ( w = W -> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) = ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
19	11, 18	mpteq12dv 4733	. . . 4 |- ( w = W -> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )
20		eqid 2622	. . . 4 |- ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) )
21		fvex 6201	. . . . 5 |- ( ( LTrn ` K ) ` W ) e. _V
22	21	mptex 6486	. . . 4 |- ( f e. ( ( LTrn ` K ) ` W ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) e. _V
23	19, 20, 22	fvmpt 6282	. . 3 |- ( W e. H -> ( ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ` W ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )
24		trlset.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
25		eqid 2622	. . . 4 |- ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) = ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) )
26	24, 25	mpteq12i 4742	. . 3 |- ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
27	23, 26	syl6eqr 2674	. 2 |- ( W e. H -> ( ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ` W ) = ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )
28	10, 27	sylan9eq 2676	1 |- ( ( K e. C /\ W e. H ) -> R = ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   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:  trlval  35449