Theorem tendoipl2

Description: Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014)

	Ref	Expression
Hypotheses	tendoicl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tendoicl.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tendoicl.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	tendoicl.i	⊢ 𝐼 = ( 𝑠 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ◡ ( 𝑠 ‘ 𝑓 ) ) )
	tendoi.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	tendoi.p	⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
	tendoi.o	⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
Assertion	tendoipl2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 𝑃 ( 𝐼 ‘ 𝑆 ) ) = 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		tendoicl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tendoicl.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		tendoicl.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		tendoicl.i	⊢ 𝐼 = ( 𝑠 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ◡ ( 𝑠 ‘ 𝑓 ) ) )
5		tendoi.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
6		tendoi.p	⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
7		tendoi.o	⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
8	1 2 3 4	tendoicl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝐼 ‘ 𝑆 ) ∈ 𝐸 )
9	1 2 3 6	tendoplcom	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐼 ‘ 𝑆 ) ∈ 𝐸 ) → ( 𝑆 𝑃 ( 𝐼 ‘ 𝑆 ) ) = ( ( 𝐼 ‘ 𝑆 ) 𝑃 𝑆 ) )
10	8 9	mpd3an3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 𝑃 ( 𝐼 ‘ 𝑆 ) ) = ( ( 𝐼 ‘ 𝑆 ) 𝑃 𝑆 ) )
11	1 2 3 4 5 6 7	tendoipl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( ( 𝐼 ‘ 𝑆 ) 𝑃 𝑆 ) = 𝑂 )
12	10 11	eqtrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 𝑃 ( 𝐼 ‘ 𝑆 ) ) = 𝑂 )