Theorem dian0

Description: The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014)

	Ref	Expression
Hypotheses	dian0.b	|- B = ( Base ` K )
	dian0.l	|- .<_ = ( le ` K )
	dian0.h	|- H = ( LHyp ` K )
	dian0.i	|- I = ( ( DIsoA ` K ) ` W )
Assertion	dian0	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) =/= (/) )

Proof

Step	Hyp	Ref	Expression
1		dian0.b	|- B = ( Base ` K )
2		dian0.l	|- .<_ = ( le ` K )
3		dian0.h	|- H = ( LHyp ` K )
4		dian0.i	|- I = ( ( DIsoA ` K ) ` W )
5		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
6	1 3 5	idltrn	|- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) )
7	6	adantr	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) )
8		eqid	|- ( 0. ` K ) = ( 0. ` K )
9		eqid	|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
10	1 8 3 9	trlid0	|- ( ( K e. HL /\ W e. H ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) = ( 0. ` K ) )
11	10	adantr	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) = ( 0. ` K ) )
12		hlatl	|- ( K e. HL -> K e. AtLat )
13	12	adantr	|- ( ( K e. HL /\ W e. H ) -> K e. AtLat )
14		simpl	|- ( ( X e. B /\ X .<_ W ) -> X e. B )
15	1 2 8	atl0le	|- ( ( K e. AtLat /\ X e. B ) -> ( 0. ` K ) .<_ X )
16	13 14 15	syl2an	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( 0. ` K ) .<_ X )
17	11 16	eqbrtrd	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) .<_ X )
18	1 2 3 5 9 4	diaelval	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( _I |` B ) e. ( I ` X ) <-> ( ( _I |` B ) e. ( ( LTrn ` K ) ` W ) /\ ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) .<_ X ) ) )
19	7 17 18	mpbir2and	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( _I |` B ) e. ( I ` X ) )
20	19	ne0d	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) =/= (/) )