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Metamath Proof Explorer

Theorem dibn0

Description: The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014)

Ref Expression
Hypotheses dibn0.b 
|- B = ( Base ` K )
dibn0.l 
|- .<_ = ( le ` K )
dibn0.h 
|- H = ( LHyp ` K )
dibn0.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibn0 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) =/= (/) )

Proof

Step Hyp Ref Expression
1 dibn0.b 
 |-  B = ( Base ` K )
2 dibn0.l 
 |-  .<_ = ( le ` K )
3 dibn0.h 
 |-  H = ( LHyp ` K )
4 dibn0.i 
 |-  I = ( ( DIsoB ` K ) ` W )
5 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
6 eqid 
 |-  ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
7 eqid 
 |-  ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
8 1 2 3 5 6 7 4 dibval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) )
9 1 2 3 7 dian0 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( DIsoA ` K ) ` W ) ` X ) =/= (/) )
10 fvex 
 |-  ( ( LTrn ` K ) ` W ) e. _V
11 10 mptex 
 |-  ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) e. _V
12 11 snnz 
 |-  { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } =/= (/)
13 9 12 jctir 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( ( DIsoA ` K ) ` W ) ` X ) =/= (/) /\ { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } =/= (/) ) )
14 xpnz 
 |-  ( ( ( ( ( DIsoA ` K ) ` W ) ` X ) =/= (/) /\ { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } =/= (/) ) <-> ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) =/= (/) )
15 13 14 sylib 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) =/= (/) )
16 8 15 eqnetrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) =/= (/) )