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Metamath Proof Explorer

Theorem djhexmid

Description: Excluded middle property of DVecH vector space closed subspace join. (Contributed by NM, 22-Jul-2014)

Ref Expression
Hypotheses djhexmid.h 
|- H = ( LHyp ` K )
djhexmid.u 
|- U = ( ( DVecH ` K ) ` W )
djhexmid.v 
|- V = ( Base ` U )
djhexmid.o 
|- ._|_ = ( ( ocH ` K ) ` W )
djhexmid.j 
|- .\/ = ( ( joinH ` K ) ` W )
Assertion djhexmid 
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( X .\/ ( ._|_ ` X ) ) = V )

Proof

Step Hyp Ref Expression
1 djhexmid.h 
 |-  H = ( LHyp ` K )
2 djhexmid.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 djhexmid.v 
 |-  V = ( Base ` U )
4 djhexmid.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
5 djhexmid.j 
 |-  .\/ = ( ( joinH ` K ) ` W )
6 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( K e. HL /\ W e. H ) )
7 simpr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> X C_ V )
8 1 2 3 4 dochssv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) C_ V )
9 1 2 3 4 5 djhval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X C_ V /\ ( ._|_ ` X ) C_ V ) ) -> ( X .\/ ( ._|_ ` X ) ) = ( ._|_ ` ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) ) )
10 6 7 8 9 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( X .\/ ( ._|_ ` X ) ) = ( ._|_ ` ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) ) )
11 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
12 1 2 3 11 4 dochlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) e. ( LSubSp ` U ) )
13 eqid 
 |-  ( 0g ` U ) = ( 0g ` U )
14 1 2 11 13 4 dochnoncon 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` X ) e. ( LSubSp ` U ) ) -> ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) = { ( 0g ` U ) } )
15 12 14 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) = { ( 0g ` U ) } )
16 1 2 4 3 13 doch1 
 |-  ( ( K e. HL /\ W e. H ) -> ( ._|_ ` V ) = { ( 0g ` U ) } )
17 16 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` V ) = { ( 0g ` U ) } )
18 15 17 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) = ( ._|_ ` V ) )
19 18 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) ) = ( ._|_ ` ( ._|_ ` V ) ) )
20 eqid 
 |-  ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
21 1 20 2 3 dih1rn 
 |-  ( ( K e. HL /\ W e. H ) -> V e. ran ( ( DIsoH ` K ) ` W ) )
22 21 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> V e. ran ( ( DIsoH ` K ) ` W ) )
23 1 20 4 dochoc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ V e. ran ( ( DIsoH ` K ) ` W ) ) -> ( ._|_ ` ( ._|_ ` V ) ) = V )
24 22 23 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` ( ._|_ ` V ) ) = V )
25 10 19 24 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( X .\/ ( ._|_ ` X ) ) = V )