Metamath Proof Explorer


Theorem hhsst

Description: A member of SH is a subspace. (Contributed by NM, 6-Apr-2008) (New usage is discouraged.)

Ref Expression
Hypotheses hhsst.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
hhsst.2 𝑊 = ⟨ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ , ( norm𝐻 ) ⟩
Assertion hhsst ( 𝐻S𝑊 ∈ ( SubSp ‘ 𝑈 ) )

Proof

Step Hyp Ref Expression
1 hhsst.1 𝑈 = ⟨ ⟨ + , · ⟩ , norm
2 hhsst.2 𝑊 = ⟨ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ , ( norm𝐻 ) ⟩
3 2 hhssnvt ( 𝐻S𝑊 ∈ NrmCVec )
4 resss ( + ↾ ( 𝐻 × 𝐻 ) ) ⊆ +
5 resss ( · ↾ ( ℂ × 𝐻 ) ) ⊆ ·
6 resss ( norm𝐻 ) ⊆ norm
7 4 5 6 3pm3.2i ( ( + ↾ ( 𝐻 × 𝐻 ) ) ⊆ + ∧ ( · ↾ ( ℂ × 𝐻 ) ) ⊆ · ∧ ( norm𝐻 ) ⊆ norm )
8 3 7 jctir ( 𝐻S → ( 𝑊 ∈ NrmCVec ∧ ( ( + ↾ ( 𝐻 × 𝐻 ) ) ⊆ + ∧ ( · ↾ ( ℂ × 𝐻 ) ) ⊆ · ∧ ( norm𝐻 ) ⊆ norm ) ) )
9 1 hhnv 𝑈 ∈ NrmCVec
10 1 hhva + = ( +𝑣𝑈 )
11 2 hhssva ( + ↾ ( 𝐻 × 𝐻 ) ) = ( +𝑣𝑊 )
12 1 hhsm · = ( ·𝑠OLD𝑈 )
13 2 hhsssm ( · ↾ ( ℂ × 𝐻 ) ) = ( ·𝑠OLD𝑊 )
14 1 hhnm norm = ( normCV𝑈 )
15 2 hhssnm ( norm𝐻 ) = ( normCV𝑊 )
16 eqid ( SubSp ‘ 𝑈 ) = ( SubSp ‘ 𝑈 )
17 10 11 12 13 14 15 16 isssp ( 𝑈 ∈ NrmCVec → ( 𝑊 ∈ ( SubSp ‘ 𝑈 ) ↔ ( 𝑊 ∈ NrmCVec ∧ ( ( + ↾ ( 𝐻 × 𝐻 ) ) ⊆ + ∧ ( · ↾ ( ℂ × 𝐻 ) ) ⊆ · ∧ ( norm𝐻 ) ⊆ norm ) ) ) )
18 9 17 ax-mp ( 𝑊 ∈ ( SubSp ‘ 𝑈 ) ↔ ( 𝑊 ∈ NrmCVec ∧ ( ( + ↾ ( 𝐻 × 𝐻 ) ) ⊆ + ∧ ( · ↾ ( ℂ × 𝐻 ) ) ⊆ · ∧ ( norm𝐻 ) ⊆ norm ) ) )
19 8 18 sylibr ( 𝐻S𝑊 ∈ ( SubSp ‘ 𝑈 ) )