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 ⊢ ≤ ˙ = ≤ 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 ⊢ ≤ ˙ = ≤ 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