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Metamath Proof Explorer

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 )