Theorem tendospdi1

Description: Forward distributive law for 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	tendospdi1	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑈 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑈 ‘ 𝐺 ) ) )

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		simpll	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐾 ∈ 𝑉 )
5		simplr	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝑊 ∈ 𝐻 )
6		simpr1	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝑈 ∈ 𝐸 )
7		simpr2	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐹 ∈ 𝑇 )
8		simpr3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐺 ∈ 𝑇 )
9	1 2 3	tendovalco	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑈 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑈 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑈 ‘ 𝐺 ) ) )
10	4 5 6 7 8 9	syl32anc	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑈 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑈 ‘ 𝐺 ) ) )