Metamath Proof Explorer

Theorem trljat2

Description: The value of a translation of an atom P not under the fiducial co-atom W , joined with trace. Equation above Lemma C in Crawley p. 112. (Contributed by NM, 25-May-2012)

Ref Expression
Hypotheses trljat.l ⊢ ≤ = ( le ‘ 𝐾 )
trljat.j ⊢ ∨ = ( join ‘ 𝐾 )
trljat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
trljat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trljat.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trljat.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trljat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )

Proof

Step Hyp Ref Expression
1 trljat.l ⊢ ≤ = ( le ‘ 𝐾 )
2 trljat.j ⊢ ∨ = ( join ‘ 𝐾 )
3 trljat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 trljat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 trljat.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 trljat.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
8 1 3 4 5 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
9 8 3adant3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
10 7 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
11 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 )
12 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
13 12 3 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
14 11 13 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
15 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
16 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
17 12 4 5 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
18 15 16 14 17 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
19 12 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
20 10 14 18 19 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
21 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 )
22 12 4 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
23 21 22 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
24 12 1 2 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
25 10 14 18 24 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
26 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
27 12 1 2 26 3 atmod2i1 ⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) ) )
28 7 9 20 23 25 27 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) ) )
29 1 3 4 5 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
30 eqid ⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
31 1 2 30 3 4 lhpjat1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 1. ‘ 𝐾 ) )
32 7 21 29 31 syl21anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 1. ‘ 𝐾 ) )
33 32 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) )
34 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
35 7 34 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ OL )
36 12 26 30 olm11 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
37 35 20 36 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
38 28 33 37 3eqtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
39 1 2 26 3 4 5 6 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
40 39 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
41 12 4 5 6 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
42 15 16 41 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
43 12 2 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) )
44 10 42 18 43 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) )
45 38 40 44 3eqtr2rd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )