Metamath Proof Explorer

Theorem idltrn

Description: The identity function is a lattice translation. Remark below Lemma B in Crawley p. 112. (Contributed by NM, 18-May-2012)

Ref Expression
Hypotheses idltrn.b ⊢ B = Base K
idltrn.h ⊢ H = LHyp ⁡ K
idltrn.t ⊢ T = LTrn ⁡ K ⁡ W
Assertion idltrn ⊢ K ∈ HL ∧ W ∈ H → I ↾ B ∈ T

Proof

Step Hyp Ref Expression
1 idltrn.b ⊢ B = Base K
2 idltrn.h ⊢ H = LHyp ⁡ K
3 idltrn.t ⊢ T = LTrn ⁡ K ⁡ W
4 eqid ⊢ LDil ⁡ K ⁡ W = LDil ⁡ K ⁡ W
5 1 2 4 idldil ⊢ K ∈ HL ∧ W ∈ H → I ↾ B ∈ LDil ⁡ K ⁡ W
6 simpll ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → K ∈ HL ∧ W ∈ H
7 simplrr ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → q ∈ Atoms ⁡ K
8 simprr ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → ¬ q ≤ K W
9 eqid ⊢ ≤ K = ≤ K
10 eqid ⊢ meet ⁡ K = meet ⁡ K
11 eqid ⊢ 0. ⁡ K = 0. ⁡ K
12 eqid ⊢ Atoms ⁡ K = Atoms ⁡ K
13 9 10 11 12 2 lhpmat ⊢ K ∈ HL ∧ W ∈ H ∧ q ∈ Atoms ⁡ K ∧ ¬ q ≤ K W → q meet ⁡ K W = 0. ⁡ K
14 6 7 8 13 syl12anc ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → q meet ⁡ K W = 0. ⁡ K
15 1 12 atbase ⊢ q ∈ Atoms ⁡ K → q ∈ B
16 fvresi ⊢ q ∈ B → I ↾ B ⁡ q = q
17 7 15 16 3syl ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → I ↾ B ⁡ q = q
18 17 oveq2d ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → q join ⁡ K I ↾ B ⁡ q = q join ⁡ K q
19 simplll ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → K ∈ HL
20 eqid ⊢ join ⁡ K = join ⁡ K
21 20 12 hlatjidm ⊢ K ∈ HL ∧ q ∈ Atoms ⁡ K → q join ⁡ K q = q
22 19 7 21 syl2anc ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → q join ⁡ K q = q
23 18 22 eqtrd ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → q join ⁡ K I ↾ B ⁡ q = q
24 23 oveq1d ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → q join ⁡ K I ↾ B ⁡ q meet ⁡ K W = q meet ⁡ K W
25 simplrl ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → p ∈ Atoms ⁡ K
26 1 12 atbase ⊢ p ∈ Atoms ⁡ K → p ∈ B
27 fvresi ⊢ p ∈ B → I ↾ B ⁡ p = p
28 25 26 27 3syl ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → I ↾ B ⁡ p = p
29 28 oveq2d ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K I ↾ B ⁡ p = p join ⁡ K p
30 20 12 hlatjidm ⊢ K ∈ HL ∧ p ∈ Atoms ⁡ K → p join ⁡ K p = p
31 19 25 30 syl2anc ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K p = p
32 29 31 eqtrd ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K I ↾ B ⁡ p = p
33 32 oveq1d ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K I ↾ B ⁡ p meet ⁡ K W = p meet ⁡ K W
34 simprl ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → ¬ p ≤ K W
35 9 10 11 12 2 lhpmat ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ ¬ p ≤ K W → p meet ⁡ K W = 0. ⁡ K
36 6 25 34 35 syl12anc ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → p meet ⁡ K W = 0. ⁡ K
37 33 36 eqtrd ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K I ↾ B ⁡ p meet ⁡ K W = 0. ⁡ K
38 14 24 37 3eqtr4rd ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K ∧ ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K I ↾ B ⁡ p meet ⁡ K W = q join ⁡ K I ↾ B ⁡ q meet ⁡ K W
39 38 ex ⊢ K ∈ HL ∧ W ∈ H ∧ p ∈ Atoms ⁡ K ∧ q ∈ Atoms ⁡ K → ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K I ↾ B ⁡ p meet ⁡ K W = q join ⁡ K I ↾ B ⁡ q meet ⁡ K W
40 39 ralrimivva ⊢ K ∈ HL ∧ W ∈ H → ∀ p ∈ Atoms ⁡ K ∀ q ∈ Atoms ⁡ K ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K I ↾ B ⁡ p meet ⁡ K W = q join ⁡ K I ↾ B ⁡ q meet ⁡ K W
41 9 20 10 12 2 4 3 isltrn ⊢ K ∈ HL ∧ W ∈ H → I ↾ B ∈ T ↔ I ↾ B ∈ LDil ⁡ K ⁡ W ∧ ∀ p ∈ Atoms ⁡ K ∀ q ∈ Atoms ⁡ K ¬ p ≤ K W ∧ ¬ q ≤ K W → p join ⁡ K I ↾ B ⁡ p meet ⁡ K W = q join ⁡ K I ↾ B ⁡ q meet ⁡ K W
42 5 40 41 mpbir2and ⊢ K ∈ HL ∧ W ∈ H → I ↾ B ∈ T