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 ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
djhunss.x ( 𝜑𝑋𝑉 )
djhunss.y ( 𝜑𝑌𝑉 )
Assertion djhunssN ( 𝜑 → ( 𝑋𝑌 ) ⊆ ( 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 ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
6 djhunss.x ( 𝜑𝑋𝑉 )
7 djhunss.y ( 𝜑𝑌𝑉 )
8 1 2 5 dvhlmod ( 𝜑𝑈 ∈ LMod )
9 6 7 unssd ( 𝜑 → ( 𝑋𝑌 ) ⊆ 𝑉 )
10 eqid ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
11 3 10 lspssid ( ( 𝑈 ∈ LMod ∧ ( 𝑋𝑌 ) ⊆ 𝑉 ) → ( 𝑋𝑌 ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋𝑌 ) ) )
12 8 9 11 syl2anc ( 𝜑 → ( 𝑋𝑌 ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋𝑌 ) ) )
13 1 2 3 10 4 5 6 7 djhspss ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋𝑌 ) ) ⊆ ( 𝑋 𝑌 ) )
14 12 13 sstrd ( 𝜑 → ( 𝑋𝑌 ) ⊆ ( 𝑋 𝑌 ) )