Theorem cdlemftr0

Description: Special case of cdlemf showing existence of a non-identity translation. (Contributed by NM, 1-Aug-2013)

	Ref	Expression
Hypotheses	cdlemftr0.b	|- B = ( Base ` K )
	cdlemftr0.h	|- H = ( LHyp ` K )
	cdlemftr0.t	|- T = ( ( LTrn ` K ) ` W )
Assertion	cdlemftr0	|- ( ( K e. HL /\ W e. H ) -> E. f e. T f =/= ( _I |` B ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemftr0.b	|- B = ( Base ` K )
2		cdlemftr0.h	|- H = ( LHyp ` K )
3		cdlemftr0.t	|- T = ( ( LTrn ` K ) ` W )
4		eqid	|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
5	1 2 3 4	cdlemftr1	|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( ( ( trL ` K ) ` W ) ` f ) =/= _I ) )
6		simpl	|- ( ( f =/= ( _I |` B ) /\ ( ( ( trL ` K ) ` W ) ` f ) =/= _I ) -> f =/= ( _I |` B ) )
7	6	reximi	|- ( E. f e. T ( f =/= ( _I |` B ) /\ ( ( ( trL ` K ) ` W ) ` f ) =/= _I ) -> E. f e. T f =/= ( _I |` B ) )
8	5 7	syl	|- ( ( K e. HL /\ W e. H ) -> E. f e. T f =/= ( _I |` B ) )