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Metamath Proof Explorer

Theorem tendotp

Description: Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013)

Ref Expression
Hypotheses tendoset.l ⊢ ≤ = ( le ‘ 𝐾 )
tendoset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
tendoset.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
tendoset.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
tendoset.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion tendotp ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 tendoset.l ⊢ ≤ = ( le ‘ 𝐾 )
2 tendoset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 tendoset.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 tendoset.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5 tendoset.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
6 1 2 3 4 5 istendo ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 ↔ ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ) )
7 2fveq3 ⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) = ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) )
8 fveq2 ⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐹 ) )
9 7 8 breq12d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ↔ ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) )
10 9 rspccv ⊢ ( ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) )
11 10 3ad2ant3 ⊢ ( ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) )
12 6 11 biimtrdi ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ) )
13 12 3imp ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) )