Theorem cdlemftr2

Description: Special case of cdlemf showing existence of non-identity translation with trace different from any 2 given lattice elements. (Contributed by NM, 25-Jul-2013)

	Ref	Expression
Hypotheses	cdlemftr.b	|- B = ( Base ` K )
	cdlemftr.h	|- H = ( LHyp ` K )
	cdlemftr.t	|- T = ( ( LTrn ` K ) ` W )
	cdlemftr.r	|- R = ( ( trL ` K ) ` W )
Assertion	cdlemftr2	|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X /\ ( R ` f ) =/= Y ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemftr.b	|- B = ( Base ` K )
2		cdlemftr.h	|- H = ( LHyp ` K )
3		cdlemftr.t	|- T = ( ( LTrn ` K ) ` W )
4		cdlemftr.r	|- R = ( ( trL ` K ) ` W )
5	1 2 3 4	cdlemftr3	|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Y ) ) )
6		simpl	|- ( ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Y ) ) -> f =/= ( _I |` B ) )
7		simpr1	|- ( ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Y ) ) -> ( R ` f ) =/= X )
8		simpr2	|- ( ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Y ) ) -> ( R ` f ) =/= Y )
9	6 7 8	3jca	|- ( ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Y ) ) -> ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X /\ ( R ` f ) =/= Y ) )
10	9	reximi	|- ( E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Y ) ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X /\ ( R ` f ) =/= Y ) )
11	5 10	syl	|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X /\ ( R ` f ) =/= Y ) )