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Metamath Proof Explorer

Theorem sspnv

Description: A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008) (New usage is discouraged.)

Ref Expression
Hypothesis sspnv.h ⊢ 𝐻 = ( SubSp ‘ 𝑈 )
Assertion sspnv ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ NrmCVec )

Proof

Step Hyp Ref Expression
1 sspnv.h ⊢ 𝐻 = ( SubSp ‘ 𝑈 )
2 eqid ⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 )
3 eqid ⊢ ( +𝑣 ‘ 𝑊 ) = ( +𝑣 ‘ 𝑊 )
4 eqid ⊢ ( ·𝑠OLD ‘ 𝑈 ) = ( ·𝑠OLD ‘ 𝑈 )
5 eqid ⊢ ( ·𝑠OLD ‘ 𝑊 ) = ( ·𝑠OLD ‘ 𝑊 )
6 eqid ⊢ ( normCV ‘ 𝑈 ) = ( normCV ‘ 𝑈 )
7 eqid ⊢ ( normCV ‘ 𝑊 ) = ( normCV ‘ 𝑊 )
8 2 3 4 5 6 7 1 isssp ⊢ ( 𝑈 ∈ NrmCVec → ( 𝑊 ∈ 𝐻 ↔ ( 𝑊 ∈ NrmCVec ∧ ( ( +𝑣 ‘ 𝑊 ) ⊆ ( +𝑣 ‘ 𝑈 ) ∧ ( ·𝑠OLD ‘ 𝑊 ) ⊆ ( ·𝑠OLD ‘ 𝑈 ) ∧ ( normCV ‘ 𝑊 ) ⊆ ( normCV ‘ 𝑈 ) ) ) ) )
9 8 simprbda ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ NrmCVec )