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Metamath Proof Explorer

Theorem tendocoval

Description: Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013)

Ref Expression
Hypotheses tendof.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
tendof.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
tendof.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion tendocoval ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 ∘ 𝑉 ) ‘ 𝐹 ) = ( 𝑈 ‘ ( 𝑉 ‘ 𝐹 ) ) )

Proof

Step Hyp Ref Expression
1 tendof.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 tendof.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 tendof.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 simp1 ⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) )
5 simp2r ⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑉 ∈ 𝐸 )
6 1 2 3 tendof ⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐸 ) → 𝑉 : 𝑇 ⟶ 𝑇 )
7 4 5 6 syl2anc ⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑉 : 𝑇 ⟶ 𝑇 )
8 simp3 ⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 )
9 fvco3 ⊢ ( ( 𝑉 : 𝑇 ⟶ 𝑇 ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 ∘ 𝑉 ) ‘ 𝐹 ) = ( 𝑈 ‘ ( 𝑉 ‘ 𝐹 ) ) )
10 7 8 9 syl2anc ⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 ∘ 𝑉 ) ‘ 𝐹 ) = ( 𝑈 ‘ ( 𝑉 ‘ 𝐹 ) ) )