Theorem trlval 35449

Description: The value of the trace of a lattice translation. (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
trlval	|- ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> ( R ` F ) = ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( F ` p ) ) ./\ W ) ) ) )

Distinct variable groups:   A, p   x, B   x, p, K   W, p, x   F, p, x

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

Proof of Theorem trlval

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlset.b	. . . 4 |- B = ( Base ` K )
2		trlset.l	. . . 4 |- .<_ = ( le ` K )
3		trlset.j	. . . 4 |- .\/ = ( join ` K )
4		trlset.m	. . . 4 |- ./\ = ( meet ` K )
5		trlset.a	. . . 4 |- A = ( Atoms ` K )
6		trlset.h	. . . 4 |- H = ( LHyp ` K )
7		trlset.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
8		trlset.r	. . . 4 |- R = ( ( trL ` K ) ` W )
9	1, 2, 3, 4, 5, 6, 7, 8	trlset 35448	. . 3 |- ( ( K e. V /\ W e. H ) -> R = ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )
10	9	fveq1d 6193	. 2 |- ( ( K e. V /\ W e. H ) -> ( R ` F ) = ( ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) ` F ) )
11		fveq1 6190	. . . . . . . . 9 |- ( f = F -> ( f ` p ) = ( F ` p ) )
12	11	oveq2d 6666	. . . . . . . 8 |- ( f = F -> ( p .\/ ( f ` p ) ) = ( p .\/ ( F ` p ) ) )
13	12	oveq1d 6665	. . . . . . 7 |- ( f = F -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( p .\/ ( F ` p ) ) ./\ W ) )
14	13	eqeq2d 2632	. . . . . 6 |- ( f = F -> ( x = ( ( p .\/ ( f ` p ) ) ./\ W ) <-> x = ( ( p .\/ ( F ` p ) ) ./\ W ) ) )
15	14	imbi2d 330	. . . . 5 |- ( f = F -> ( ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) <-> ( -. p .<_ W -> x = ( ( p .\/ ( F ` p ) ) ./\ W ) ) ) )
16	15	ralbidv 2986	. . . 4 |- ( f = F -> ( A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) <-> A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( F ` p ) ) ./\ W ) ) ) )
17	16	riotabidv 6613	. . 3 |- ( f = F -> ( 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 ) ) ) )
18		eqid 2622	. . 3 |- ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) = ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
19		riotaex 6615	. . 3 |- ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( F ` p ) ) ./\ W ) ) ) e. _V
20	17, 18, 19	fvmpt 6282	. 2 |- ( F e. T -> ( ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) ` F ) = ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( F ` p ) ) ./\ W ) ) ) )
21	10, 20	sylan9eq 2676	1 |- ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> ( R ` F ) = ( 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:  trlval2  35450