Theorem tendo0cl

Description: The additive identity is a trace-preserving endormorphism. (Contributed by NM, 12-Jun-2013)

	Ref	Expression
Hypotheses	tendo0.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	tendo0.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tendo0.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tendo0.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	tendo0.o	⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
Assertion	tendo0cl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		tendo0.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		tendo0.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		tendo0.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4		tendo0.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5		tendo0.o	⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
6		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7		eqid	⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8		id	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9	1 2 3	idltrn	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝐵 ) ∈ 𝑇 )
10	9	adantr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ) → ( I ↾ 𝐵 ) ∈ 𝑇 )
11	5	tendo0cbv	⊢ 𝑂 = ( 𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
12	10 11	fmptd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 : 𝑇 ⟶ 𝑇 )
13	1 2 3 4 5	tendo0co2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ ℎ ∈ 𝑇 ) → ( 𝑂 ‘ ( 𝑔 ∘ ℎ ) ) = ( ( 𝑂 ‘ 𝑔 ) ∘ ( 𝑂 ‘ ℎ ) ) )
14	1 2 3 4 5 6 7	tendo0tp	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑂 ‘ 𝑔 ) ) ( le ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑔 ) )
15	6 2 3 7 4 8 12 13 14	istendod	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 )