Metamath Proof Explorer

Theorem trljat1

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. TODO: shorten with atmod3i1 ? (Contributed by NM, 22-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 trljat1 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 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 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8 1 2 7 3 4 5 6 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
9 8 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ 𝑃 ) = ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ 𝑃 ) )
10 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
11 10 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
12 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 )
13 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
14 13 3 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
15 12 14 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
16 13 4 5 6 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
17 16 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
18 13 2 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝑅 ‘ 𝐹 ) ∨ 𝑃 ) )
19 11 15 17 18 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( ( 𝑅 ‘ 𝐹 ) ∨ 𝑃 ) )
20 13 4 5 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
21 15 20 syld3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
22 13 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
23 11 15 21 22 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
24 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 )
25 13 4 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
26 24 25 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
27 13 1 2 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → 𝑃 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
28 11 15 21 27 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
29 13 1 2 7 3 atmod2i1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑃 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ 𝑃 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ 𝑃 ) ) )
30 10 12 23 26 28 29 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ 𝑃 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ 𝑃 ) ) )
31 eqid ⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
32 1 2 31 3 4 lhpjat1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑊 ∨ 𝑃 ) = ( 1. ‘ 𝐾 ) )
33 32 3adant2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑊 ∨ 𝑃 ) = ( 1. ‘ 𝐾 ) )
34 33 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ 𝑃 ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) )
35 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
36 10 35 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ OL )
37 13 7 31 olm11 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
38 36 23 37 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
39 30 34 38 3eqtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ 𝑃 ) )
40 9 19 39 3eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )