Metamath Proof Explorer

Theorem dicn0

Description: The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014)

Ref Expression
Hypotheses dicn0.l 
|- .<_ = ( le ` K )
dicn0.a 
|- A = ( Atoms ` K )
dicn0.h 
|- H = ( LHyp ` K )
dicn0.i 
|- I = ( ( DIsoC ` K ) ` W )
Assertion dicn0 
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) =/= (/) )

Proof

Step Hyp Ref Expression
1 dicn0.l 
 |-  .<_ = ( le ` K )
2 dicn0.a 
 |-  A = ( Atoms ` K )
3 dicn0.h 
 |-  H = ( LHyp ` K )
4 dicn0.i 
 |-  I = ( ( DIsoC ` K ) ` W )
5 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( K e. HL /\ W e. H ) )
6 eqid 
 |-  ( oc ` K ) = ( oc ` K )
7 1 6 2 3 lhpocnel 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
8 7 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
9 simpr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Q e. A /\ -. Q .<_ W ) )
10 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11 eqid 
 |-  ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q )
12 1 2 3 10 11 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) )
13 5 8 9 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) )
14 eqid 
 |-  ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 14 15 tendo02 
 |-  ( ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) -> ( ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( _I |` ( Base ` K ) ) )
17 13 16 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( _I |` ( Base ` K ) ) )
18 17 eqcomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( _I |` ( Base ` K ) ) = ( ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) )
19 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
20 15 3 10 19 14 tendo0cl 
 |-  ( ( K e. HL /\ W e. H ) -> ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) e. ( ( TEndo ` K ) ` W ) )
21 20 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) e. ( ( TEndo ` K ) ` W ) )
22 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
23 fvex 
 |-  ( Base ` K ) e. _V
24 resiexg 
 |-  ( ( Base ` K ) e. _V -> ( _I |` ( Base ` K ) ) e. _V )
25 23 24 ax-mp 
 |-  ( _I |` ( Base ` K ) ) e. _V
26 fvex 
 |-  ( ( LTrn ` K ) ` W ) e. _V
27 26 mptex 
 |-  ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) e. _V
28 1 2 3 22 10 19 4 25 27 dicopelval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. ( _I |` ( Base ` K ) ) , ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) >. e. ( I ` Q ) <-> ( ( _I |` ( Base ` K ) ) = ( ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) /\ ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) e. ( ( TEndo ` K ) ` W ) ) ) )
29 18 21 28 mpbir2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> <. ( _I |` ( Base ` K ) ) , ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) >. e. ( I ` Q ) )
30 29 ne0d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) =/= (/) )