Theorem tendo1mulr

Description: Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013)

	Ref	Expression
Hypotheses	tendof.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tendof.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tendof.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion	tendo1mulr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ) → ( 𝑈 ∘ ( I ↾ 𝑇 ) ) = 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		tendof.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tendof.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		tendof.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4	1 2 3	tendof	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ) → 𝑈 : 𝑇 ⟶ 𝑇 )
5		fcoi1	⊢ ( 𝑈 : 𝑇 ⟶ 𝑇 → ( 𝑈 ∘ ( I ↾ 𝑇 ) ) = 𝑈 )
6	4 5	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ) → ( 𝑈 ∘ ( I ↾ 𝑇 ) ) = 𝑈 )