This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem tendo0plr

Description: Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014)

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 ) )
tendo0pl.p ⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
Assertion tendo0plr ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 𝑃 𝑂 ) = 𝑆 )

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 tendo0pl.p ⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
7 1 2 3 4 5 tendo0cl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 )
8 7 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → 𝑂 ∈ 𝐸 )
9 2 3 4 6 tendoplcom ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝑂 ∈ 𝐸 ) → ( 𝑆 𝑃 𝑂 ) = ( 𝑂 𝑃 𝑆 ) )
10 8 9 mpd3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 𝑃 𝑂 ) = ( 𝑂 𝑃 𝑆 ) )
11 1 2 3 4 5 6 tendo0pl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑂 𝑃 𝑆 ) = 𝑆 )
12 10 11 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 𝑃 𝑂 ) = 𝑆 )