This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem trlatn0

Description: The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013)

Ref Expression
Hypotheses trl0a.z ⊢ 0 = ( 0. ‘ 𝐾 )
trl0a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
trl0a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trl0a.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trl0a.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlatn0 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ↔ ( 𝑅 ‘ 𝐹 ) ≠ 0 ) )

Proof

Step Hyp Ref Expression
1 trl0a.z ⊢ 0 = ( 0. ‘ 𝐾 )
2 trl0a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 trl0a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 trl0a.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 trl0a.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
7 6 ad3antrrr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ) → 𝐾 ∈ AtLat )
8 1 2 atn0 ⊢ ( ( 𝐾 ∈ AtLat ∧ ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ) → ( 𝑅 ‘ 𝐹 ) ≠ 0 )
9 7 8 sylancom ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ) → ( 𝑅 ‘ 𝐹 ) ≠ 0 )
10 9 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 → ( 𝑅 ‘ 𝐹 ) ≠ 0 ) )
11 1 2 3 4 5 trlator0 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ∨ ( 𝑅 ‘ 𝐹 ) = 0 ) )
12 11 ord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ¬ ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 → ( 𝑅 ‘ 𝐹 ) = 0 ) )
13 12 necon1ad ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝐹 ) ≠ 0 → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ) )
14 10 13 impbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ↔ ( 𝑅 ‘ 𝐹 ) ≠ 0 ) )