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 ⊢ ≤ ˙ = ≤ 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 ⊢ ≤ ˙ = ≤ 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