Metamath Proof Explorer


Theorem djhunssN

Description: Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014) (New usage is discouraged.)

Ref Expression
Hypotheses djhunss.h
|- H = ( LHyp ` K )
djhunss.u
|- U = ( ( DVecH ` K ) ` W )
djhunss.v
|- V = ( Base ` U )
djhunss.j
|- .\/ = ( ( joinH ` K ) ` W )
djhunss.k
|- ( ph -> ( K e. HL /\ W e. H ) )
djhunss.x
|- ( ph -> X C_ V )
djhunss.y
|- ( ph -> Y C_ V )
Assertion djhunssN
|- ( ph -> ( X u. Y ) C_ ( X .\/ Y ) )

Proof

Step Hyp Ref Expression
1 djhunss.h
 |-  H = ( LHyp ` K )
2 djhunss.u
 |-  U = ( ( DVecH ` K ) ` W )
3 djhunss.v
 |-  V = ( Base ` U )
4 djhunss.j
 |-  .\/ = ( ( joinH ` K ) ` W )
5 djhunss.k
 |-  ( ph -> ( K e. HL /\ W e. H ) )
6 djhunss.x
 |-  ( ph -> X C_ V )
7 djhunss.y
 |-  ( ph -> Y C_ V )
8 1 2 5 dvhlmod
 |-  ( ph -> U e. LMod )
9 6 7 unssd
 |-  ( ph -> ( X u. Y ) C_ V )
10 eqid
 |-  ( LSpan ` U ) = ( LSpan ` U )
11 3 10 lspssid
 |-  ( ( U e. LMod /\ ( X u. Y ) C_ V ) -> ( X u. Y ) C_ ( ( LSpan ` U ) ` ( X u. Y ) ) )
12 8 9 11 syl2anc
 |-  ( ph -> ( X u. Y ) C_ ( ( LSpan ` U ) ` ( X u. Y ) ) )
13 1 2 3 10 4 5 6 7 djhspss
 |-  ( ph -> ( ( LSpan ` U ) ` ( X u. Y ) ) C_ ( X .\/ Y ) )
14 12 13 sstrd
 |-  ( ph -> ( X u. Y ) C_ ( X .\/ Y ) )