Metamath Proof Explorer

Theorem trlat

Description: If an atom differs from its translation, the trace is an atom. Equation above Lemma C in Crawley p. 112. (Contributed by NM, 23-May-2012)

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

Proof

Step Hyp Ref Expression
1 trlat.l ⊢ ≤ = ( le ‘ 𝐾 )
2 trlat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 trlat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 trlat.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 trlat.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝐹 ∈ 𝑇 )
8 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
9 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
10 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
11 1 9 10 2 3 4 5 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
12 6 7 8 11 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
13 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑃 ∈ 𝐴 )
14 1 2 3 4 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
15 6 7 13 14 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
16 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 )
17 16 necomd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑃 ≠ ( 𝐹 ‘ 𝑃 ) )
18 1 9 10 2 3 lhpat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ 𝑃 ≠ ( 𝐹 ‘ 𝑃 ) ) ) → ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐴 )
19 6 8 15 17 18 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐴 )
20 12 19 eqeltrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )