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Metamath Proof Explorer

Theorem lss0ss

Description: The zero subspace is included in every subspace. ( sh0le analog.) (Contributed by NM, 27-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lss0cl.z ⊢ 0 = ( 0g ‘ 𝑊 )
lss0cl.s ⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
Assertion lss0ss ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) → { 0 } ⊆ 𝑋 )

Proof

Step Hyp Ref Expression
1 lss0cl.z ⊢ 0 = ( 0g ‘ 𝑊 )
2 lss0cl.s ⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
3 1 2 lss0cl ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) → 0 ∈ 𝑋 )
4 3 snssd ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑆 ) → { 0 } ⊆ 𝑋 )