Metamath Proof Explorer


Theorem ltrnu

Description: Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom W . Similar to definition of translation in Crawley p. 111. (Contributed by NM, 20-May-2012)

Ref Expression
Hypotheses ltrnu.l ˙ = K
ltrnu.j ˙ = join K
ltrnu.m ˙ = meet K
ltrnu.a A = Atoms K
ltrnu.h H = LHyp K
ltrnu.t T = LTrn K W
Assertion ltrnu K V W H F T P A ¬ P ˙ W Q A ¬ Q ˙ W P ˙ F P ˙ W = Q ˙ F Q ˙ W

Proof

Step Hyp Ref Expression
1 ltrnu.l ˙ = K
2 ltrnu.j ˙ = join K
3 ltrnu.m ˙ = meet K
4 ltrnu.a A = Atoms K
5 ltrnu.h H = LHyp K
6 ltrnu.t T = LTrn K W
7 an4 P A ¬ P ˙ W Q A ¬ Q ˙ W P A Q A ¬ P ˙ W ¬ Q ˙ W
8 simpr K V W H F T P A Q A P A Q A
9 simplr K V W H F T P A Q A F T
10 eqid LDil K W = LDil K W
11 1 2 3 4 5 10 6 isltrn K V W H F T F LDil K W p A q A ¬ p ˙ W ¬ q ˙ W p ˙ F p ˙ W = q ˙ F q ˙ W
12 11 ad2antrr K V W H F T P A Q A F T F LDil K W p A q A ¬ p ˙ W ¬ q ˙ W p ˙ F p ˙ W = q ˙ F q ˙ W
13 simpr F LDil K W p A q A ¬ p ˙ W ¬ q ˙ W p ˙ F p ˙ W = q ˙ F q ˙ W p A q A ¬ p ˙ W ¬ q ˙ W p ˙ F p ˙ W = q ˙ F q ˙ W
14 12 13 syl6bi K V W H F T P A Q A F T p A q A ¬ p ˙ W ¬ q ˙ W p ˙ F p ˙ W = q ˙ F q ˙ W
15 9 14 mpd K V W H F T P A Q A p A q A ¬ p ˙ W ¬ q ˙ W p ˙ F p ˙ W = q ˙ F q ˙ W
16 breq1 p = P p ˙ W P ˙ W
17 16 notbid p = P ¬ p ˙ W ¬ P ˙ W
18 17 anbi1d p = P ¬ p ˙ W ¬ q ˙ W ¬ P ˙ W ¬ q ˙ W
19 id p = P p = P
20 fveq2 p = P F p = F P
21 19 20 oveq12d p = P p ˙ F p = P ˙ F P
22 21 oveq1d p = P p ˙ F p ˙ W = P ˙ F P ˙ W
23 22 eqeq1d p = P p ˙ F p ˙ W = q ˙ F q ˙ W P ˙ F P ˙ W = q ˙ F q ˙ W
24 18 23 imbi12d p = P ¬ p ˙ W ¬ q ˙ W p ˙ F p ˙ W = q ˙ F q ˙ W ¬ P ˙ W ¬ q ˙ W P ˙ F P ˙ W = q ˙ F q ˙ W
25 breq1 q = Q q ˙ W Q ˙ W
26 25 notbid q = Q ¬ q ˙ W ¬ Q ˙ W
27 26 anbi2d q = Q ¬ P ˙ W ¬ q ˙ W ¬ P ˙ W ¬ Q ˙ W
28 id q = Q q = Q
29 fveq2 q = Q F q = F Q
30 28 29 oveq12d q = Q q ˙ F q = Q ˙ F Q
31 30 oveq1d q = Q q ˙ F q ˙ W = Q ˙ F Q ˙ W
32 31 eqeq2d q = Q P ˙ F P ˙ W = q ˙ F q ˙ W P ˙ F P ˙ W = Q ˙ F Q ˙ W
33 27 32 imbi12d q = Q ¬ P ˙ W ¬ q ˙ W P ˙ F P ˙ W = q ˙ F q ˙ W ¬ P ˙ W ¬ Q ˙ W P ˙ F P ˙ W = Q ˙ F Q ˙ W
34 24 33 rspc2v P A Q A p A q A ¬ p ˙ W ¬ q ˙ W p ˙ F p ˙ W = q ˙ F q ˙ W ¬ P ˙ W ¬ Q ˙ W P ˙ F P ˙ W = Q ˙ F Q ˙ W
35 8 15 34 sylc K V W H F T P A Q A ¬ P ˙ W ¬ Q ˙ W P ˙ F P ˙ W = Q ˙ F Q ˙ W
36 35 impr K V W H F T P A Q A ¬ P ˙ W ¬ Q ˙ W P ˙ F P ˙ W = Q ˙ F Q ˙ W
37 7 36 sylan2b K V W H F T P A ¬ P ˙ W Q A ¬ Q ˙ W P ˙ F P ˙ W = Q ˙ F Q ˙ W
38 37 3impb K V W H F T P A ¬ P ˙ W Q A ¬ Q ˙ W P ˙ F P ˙ W = Q ˙ F Q ˙ W