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Metamath Proof Explorer

Theorem tendo0tp

Description: Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013)

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 ) )
tendo0tp.l ⊢ ≤ = ( le ‘ 𝐾 )
tendo0tp.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion tendo0tp ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑂 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) )

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 tendo0tp.l ⊢ ≤ = ( le ‘ 𝐾 )
7 tendo0tp.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 5 1 tendo02 ⊢ ( 𝐹 ∈ 𝑇 → ( 𝑂 ‘ 𝐹 ) = ( I ↾ 𝐵 ) )
9 8 adantl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑂 ‘ 𝐹 ) = ( I ↾ 𝐵 ) )
10 9 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑂 ‘ 𝐹 ) ) = ( 𝑅 ‘ ( I ↾ 𝐵 ) ) )
11 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
12 1 11 2 7 trlid0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 ‘ ( I ↾ 𝐵 ) ) = ( 0. ‘ 𝐾 ) )
13 12 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( I ↾ 𝐵 ) ) = ( 0. ‘ 𝐾 ) )
14 10 13 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑂 ‘ 𝐹 ) ) = ( 0. ‘ 𝐾 ) )
15 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
16 15 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ OP )
17 1 2 3 7 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 )
18 1 6 11 op0le ⊢ ( ( 𝐾 ∈ OP ∧ ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 ) → ( 0. ‘ 𝐾 ) ≤ ( 𝑅 ‘ 𝐹 ) )
19 16 17 18 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 0. ‘ 𝐾 ) ≤ ( 𝑅 ‘ 𝐹 ) )
20 14 19 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑂 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) )