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		ffvelrn	⊢ ( ( 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 )