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 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
tendo0pl.p 𝑃 = ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑠𝑓 ) ∘ ( 𝑡𝑓 ) ) ) )
Assertion tendo0plr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ) → ( 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 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
6 tendo0pl.p 𝑃 = ( 𝑠𝐸 , 𝑡𝐸 ↦ ( 𝑓𝑇 ↦ ( ( 𝑠𝑓 ) ∘ ( 𝑡𝑓 ) ) ) )
7 1 2 3 4 5 tendo0cl ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝑂𝐸 )
8 7 adantr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ) → 𝑂𝐸 )
9 2 3 4 6 tendoplcom ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸𝑂𝐸 ) → ( 𝑆 𝑃 𝑂 ) = ( 𝑂 𝑃 𝑆 ) )
10 8 9 mpd3an3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ) → ( 𝑆 𝑃 𝑂 ) = ( 𝑂 𝑃 𝑆 ) )
11 1 2 3 4 5 6 tendo0pl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ) → ( 𝑂 𝑃 𝑆 ) = 𝑆 )
12 10 11 eqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ) → ( 𝑆 𝑃 𝑂 ) = 𝑆 )