Theorem cdlemg1ci2

Description: Any function of the form of the function constructed for cdleme is a translation. TODO: Fix comment. (Contributed by NM, 18-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg1c.l	|- .<_ = ( le ` K )
	cdlemg1c.a	|- A = ( Atoms ` K )
	cdlemg1c.h	|- H = ( LHyp ` K )
	cdlemg1c.t	|- T = ( ( LTrn ` K ) ` W )
Assertion	cdlemg1ci2	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> F e. T )

Proof

Step	Hyp	Ref	Expression
1		cdlemg1c.l	|- .<_ = ( le ` K )
2		cdlemg1c.a	|- A = ( Atoms ` K )
3		cdlemg1c.h	|- H = ( LHyp ` K )
4		cdlemg1c.t	|- T = ( ( LTrn ` K ) ` W )
5		simpr	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> F = ( iota_ f e. T ( f ` P ) = Q ) )
6		eqid	|- ( iota_ f e. T ( f ` P ) = Q ) = ( iota_ f e. T ( f ` P ) = Q )
7	1 2 3 4 6	ltrniotacl	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ f e. T ( f ` P ) = Q ) e. T )
8	7	adantr	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> ( iota_ f e. T ( f ` P ) = Q ) e. T )
9	5 8	eqeltrd	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> F e. T )