Metamath Proof Explorer


Theorem hhsssm

Description: The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008) (New usage is discouraged.)

Ref Expression
Hypothesis hhss.1 𝑊 = ⟨ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ , ( norm𝐻 ) ⟩
Assertion hhsssm ( · ↾ ( ℂ × 𝐻 ) ) = ( ·𝑠OLD𝑊 )

Proof

Step Hyp Ref Expression
1 hhss.1 𝑊 = ⟨ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ , ( norm𝐻 ) ⟩
2 eqid ( ·𝑠OLD𝑊 ) = ( ·𝑠OLD𝑊 )
3 2 smfval ( ·𝑠OLD𝑊 ) = ( 2nd ‘ ( 1st𝑊 ) )
4 1 fveq2i ( 1st𝑊 ) = ( 1st ‘ ⟨ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ , ( norm𝐻 ) ⟩ )
5 opex ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ ∈ V
6 normf norm : ℋ ⟶ ℝ
7 ax-hilex ℋ ∈ V
8 fex ( ( norm : ℋ ⟶ ℝ ∧ ℋ ∈ V ) → norm ∈ V )
9 6 7 8 mp2an norm ∈ V
10 9 resex ( norm𝐻 ) ∈ V
11 5 10 op1st ( 1st ‘ ⟨ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ , ( norm𝐻 ) ⟩ ) = ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩
12 4 11 eqtri ( 1st𝑊 ) = ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩
13 12 fveq2i ( 2nd ‘ ( 1st𝑊 ) ) = ( 2nd ‘ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ )
14 hilablo + ∈ AbelOp
15 resexg ( + ∈ AbelOp → ( + ↾ ( 𝐻 × 𝐻 ) ) ∈ V )
16 14 15 ax-mp ( + ↾ ( 𝐻 × 𝐻 ) ) ∈ V
17 hvmulex · ∈ V
18 17 resex ( · ↾ ( ℂ × 𝐻 ) ) ∈ V
19 16 18 op2nd ( 2nd ‘ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ ) = ( · ↾ ( ℂ × 𝐻 ) )
20 3 13 19 3eqtrri ( · ↾ ( ℂ × 𝐻 ) ) = ( ·𝑠OLD𝑊 )