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Metamath Proof Explorer

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 ) =/= (/) )