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Metamath Proof Explorer

Theorem dihsmatrn

Description: The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at https://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 . (Contributed by NM, 15-Jan-2015)

Ref Expression
Hypotheses dihsmatrn.h 
|- H = ( LHyp ` K )
dihsmatrn.i 
|- I = ( ( DIsoH ` K ) ` W )
dihsmatrn.u 
|- U = ( ( DVecH ` K ) ` W )
dihsmatrn.p 
|- .(+) = ( LSSum ` U )
dihsmatrn.a 
|- A = ( LSAtoms ` U )
dihsmatrn.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dihsmatrn.x 
|- ( ph -> X e. ran I )
dihsmatrn.q 
|- ( ph -> Q e. A )
Assertion dihsmatrn 
|- ( ph -> ( X .(+) Q ) e. ran I )

Proof

Step Hyp Ref Expression
1 dihsmatrn.h 
 |-  H = ( LHyp ` K )
2 dihsmatrn.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 dihsmatrn.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dihsmatrn.p 
 |-  .(+) = ( LSSum ` U )
5 dihsmatrn.a 
 |-  A = ( LSAtoms ` U )
6 dihsmatrn.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
7 dihsmatrn.x 
 |-  ( ph -> X e. ran I )
8 dihsmatrn.q 
 |-  ( ph -> Q e. A )
9 eqid 
 |-  ( ( joinH ` K ) ` W ) = ( ( joinH ` K ) ` W )
10 1 2 9 3 4 5 6 7 8 dihjat2 
 |-  ( ph -> ( X ( ( joinH ` K ) ` W ) Q ) = ( X .(+) Q ) )
11 10 eqcomd 
 |-  ( ph -> ( X .(+) Q ) = ( X ( ( joinH ` K ) ` W ) Q ) )
12 eqid 
 |-  ( Base ` U ) = ( Base ` U )
13 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
14 1 3 2 13 dihrnlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> X e. ( LSubSp ` U ) )
15 6 7 14 syl2anc 
 |-  ( ph -> X e. ( LSubSp ` U ) )
16 1 3 6 dvhlmod 
 |-  ( ph -> U e. LMod )
17 13 5 16 8 lsatlssel 
 |-  ( ph -> Q e. ( LSubSp ` U ) )
18 1 3 12 13 4 2 9 6 15 17 djhlsmcl 
 |-  ( ph -> ( ( X .(+) Q ) e. ran I <-> ( X .(+) Q ) = ( X ( ( joinH ` K ) ` W ) Q ) ) )
19 11 18 mpbird 
 |-  ( ph -> ( X .(+) Q ) e. ran I )