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Metamath Proof Explorer

Theorem trlnidat

Description: The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in Crawley p. 112. (Contributed by NM, 23-May-2012)

Ref Expression
Hypotheses trlnidat.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
trlnidat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
trlnidat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trlnidat.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlnidat.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlnidat ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 trlnidat.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 trlnidat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 trlnidat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 trlnidat.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 trlnidat.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7 1 6 2 3 4 ltrnnid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) )
8 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) ) → 𝑝 ∈ 𝐴 )
10 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) ) → ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 )
11 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) ) → 𝐹 ∈ 𝑇 )
12 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) ) → ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 )
13 6 2 3 4 5 trlat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )
14 8 9 10 11 12 13 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )
15 14 rexlimdv3a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝐹 ‘ 𝑝 ) ≠ 𝑝 ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ) )
16 7 15 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )