Metamath Proof Explorer


Theorem djhsumss

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

Ref Expression
Hypotheses djhsumss.h 𝐻 = ( LHyp ‘ 𝐾 )
djhsumss.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
djhsumss.v 𝑉 = ( Base ‘ 𝑈 )
djhsumss.p = ( LSSum ‘ 𝑈 )
djhsumss.j = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 )
djhsumss.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
djhsumss.x ( 𝜑𝑋𝑉 )
djhsumss.y ( 𝜑𝑌𝑉 )
Assertion djhsumss ( 𝜑 → ( 𝑋 𝑌 ) ⊆ ( 𝑋 𝑌 ) )

Proof

Step Hyp Ref Expression
1 djhsumss.h 𝐻 = ( LHyp ‘ 𝐾 )
2 djhsumss.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 djhsumss.v 𝑉 = ( Base ‘ 𝑈 )
4 djhsumss.p = ( LSSum ‘ 𝑈 )
5 djhsumss.j = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 )
6 djhsumss.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
7 djhsumss.x ( 𝜑𝑋𝑉 )
8 djhsumss.y ( 𝜑𝑌𝑉 )
9 eqid ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
10 1 2 6 dvhlmod ( 𝜑𝑈 ∈ LMod )
11 3 9 4 7 8 10 lsmssspx ( 𝜑 → ( 𝑋 𝑌 ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋𝑌 ) ) )
12 1 2 3 9 5 6 7 8 djhspss ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋𝑌 ) ) ⊆ ( 𝑋 𝑌 ) )
13 11 12 sstrd ( 𝜑 → ( 𝑋 𝑌 ) ⊆ ( 𝑋 𝑌 ) )