Theorem cdlemftr1

Description: Part of proof of Lemma G of Crawley p. 116, sixth line of third paragraph on p. 117: there is "a translation h, different from the identity, such that tr h =/= tr f." (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	cdlemftr1	|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X ) )

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	cdlemftr2	|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X /\ ( R ` f ) =/= X ) )
6		3simpa	|- ( ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X /\ ( R ` f ) =/= X ) -> ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X ) )
7	6	reximi	|- ( E. f e. T ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X /\ ( R ` f ) =/= X ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X ) )
8	5 7	syl	|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( R ` f ) =/= X ) )