Theorem tendospcl

Description: Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013)

	Ref	Expression
Hypotheses	tendosp.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tendosp.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tendosp.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion	tendospcl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		tendosp.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tendosp.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		tendosp.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4	1 2 3	tendocl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 )