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Metamath Proof Explorer

Theorem homcl

Description: Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion homcl ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·op 𝑇 ) ‘ 𝐵 ) ∈ ℋ )

Proof

Step Hyp Ref Expression
1 homval ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·op 𝑇 ) ‘ 𝐵 ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) )
2 ffvelcdm ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ 𝐵 ) ∈ ℋ )
3 2 anim2i ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐵 ∈ ℋ ) ) → ( 𝐴 ∈ ℂ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℋ ) )
4 3 3impb ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ∈ ℂ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℋ ) )
5 hvmulcl ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ∈ ℋ )
6 4 5 syl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ∈ ℋ )
7 1 6 eqeltrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·op 𝑇 ) ‘ 𝐵 ) ∈ ℋ )