Theorem trlnidat

Description: The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in Crawley p. 112. (Contributed by NM, 23-May-2012)

	Ref	Expression
Hypotheses	trlnidat.b	|- B = ( Base ` K )
	trlnidat.a	|- A = ( Atoms ` K )
	trlnidat.h	|- H = ( LHyp ` K )
	trlnidat.t	|- T = ( ( LTrn ` K ) ` W )
	trlnidat.r	|- R = ( ( trL ` K ) ` W )
Assertion	trlnidat	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. A )

Proof

Step	Hyp	Ref	Expression
1		trlnidat.b	|- B = ( Base ` K )
2		trlnidat.a	|- A = ( Atoms ` K )
3		trlnidat.h	|- H = ( LHyp ` K )
4		trlnidat.t	|- T = ( ( LTrn ` K ) ` W )
5		trlnidat.r	|- R = ( ( trL ` K ) ` W )
6		eqid	|- ( le ` K ) = ( le ` K )
7	1 6 2 3 4	ltrnnid	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> E. p e. A ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) )
8		simp11	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> ( K e. HL /\ W e. H ) )
9		simp2	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> p e. A )
10		simp3l	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> -. p ( le ` K ) W )
11		simp12	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> F e. T )
12		simp3r	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> ( F ` p ) =/= p )
13	6 2 3 4 5	trlat	|- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F e. T /\ ( F ` p ) =/= p ) ) -> ( R ` F ) e. A )
14	8 9 10 11 12 13	syl122anc	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> ( R ` F ) e. A )
15	14	rexlimdv3a	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( E. p e. A ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) -> ( R ` F ) e. A ) )
16	7 15	mpd	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. A )