Theorem tendoplcl2

Description: Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013)

	Ref	Expression
Hypotheses	tendopl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tendopl.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tendopl.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	tendopl.p	⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
Assertion	tendoplcl2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ∈ 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		tendopl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tendopl.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		tendopl.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		tendopl.p	⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
5	4 2	tendopl2	⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) )
6	5	3expa	⊢ ( ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) )
7	6	3adant1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) )
8		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9	1 2 3	tendocl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 )
10	9	3adant2r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 )
11	1 2 3	tendocl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 )
12	11	3adant2l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 )
13	1 2	ltrnco	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ∧ ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 ) → ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ∈ 𝑇 )
14	8 10 12 13	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ∈ 𝑇 )
15	7 14	eqeltrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ∈ 𝑇 )