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Metamath Proof Explorer

Theorem trlid0b

Description: A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013)

Ref Expression
Hypotheses trlid0b.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
trlid0b.z ⊢ 0 = ( 0. ‘ 𝐾 )
trlid0b.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trlid0b.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlid0b.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlid0b ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ 𝐹 ) = 0 ) )

Proof

Step Hyp Ref Expression
1 trlid0b.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 trlid0b.z ⊢ 0 = ( 0. ‘ 𝐾 )
3 trlid0b.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 trlid0b.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 trlid0b.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
7 1 6 3 4 5 trlnidatb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ≠ ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ 𝐹 ) ∈ ( Atoms ‘ 𝐾 ) ) )
8 2 6 3 4 5 trlatn0 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝐹 ) ∈ ( Atoms ‘ 𝐾 ) ↔ ( 𝑅 ‘ 𝐹 ) ≠ 0 ) )
9 7 8 bitrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ≠ ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ 𝐹 ) ≠ 0 ) )
10 9 necon4bid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ 𝐹 ) = 0 ) )