Metamath Proof Explorer


Theorem ltrn2ateq

Description: Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012)

Ref Expression
Hypotheses ltrn2eq.l ˙ = K
ltrn2eq.a A = Atoms K
ltrn2eq.h H = LHyp K
ltrn2eq.t T = LTrn K W
Assertion ltrn2ateq K HL W H F T P A ¬ P ˙ W Q A ¬ Q ˙ W F P = P F Q = Q

Proof

Step Hyp Ref Expression
1 ltrn2eq.l ˙ = K
2 ltrn2eq.a A = Atoms K
3 ltrn2eq.h H = LHyp K
4 ltrn2eq.t T = LTrn K W
5 eqid Base K = Base K
6 5 1 2 3 4 ltrnideq K HL W H F T P A ¬ P ˙ W F = I Base K F P = P
7 6 3adant3r3 K HL W H F T P A ¬ P ˙ W Q A ¬ Q ˙ W F = I Base K F P = P
8 5 1 2 3 4 ltrnideq K HL W H F T Q A ¬ Q ˙ W F = I Base K F Q = Q
9 8 3adant3r2 K HL W H F T P A ¬ P ˙ W Q A ¬ Q ˙ W F = I Base K F Q = Q
10 7 9 bitr3d K HL W H F T P A ¬ P ˙ W Q A ¬ Q ˙ W F P = P F Q = Q