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Metamath Proof Explorer

Theorem trlne

Description: The trace of a lattice translation is not equal to any atom not under the fiducial co-atom W . Part of proof of Lemma C in Crawley p. 112. (Contributed by NM, 25-May-2012)

Ref Expression
Hypotheses trlne.l ⊢ ≤ = ( le ‘ 𝐾 )
trlne.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
trlne.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trlne.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlne.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlne ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ≠ ( 𝑅 ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 trlne.l ⊢ ≤ = ( le ‘ 𝐾 )
2 trlne.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 trlne.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 trlne.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 trlne.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ 𝑃 ≤ 𝑊 )
7 1 3 4 5 trlle ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 )
8 7 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 )
9 breq1 ⊢ ( 𝑃 = ( 𝑅 ‘ 𝐹 ) → ( 𝑃 ≤ 𝑊 ↔ ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ) )
10 8 9 syl5ibrcom ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 = ( 𝑅 ‘ 𝐹 ) → 𝑃 ≤ 𝑊 ) )
11 10 necon3bd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ¬ 𝑃 ≤ 𝑊 → 𝑃 ≠ ( 𝑅 ‘ 𝐹 ) ) )
12 6 11 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ≠ ( 𝑅 ‘ 𝐹 ) )