Metamath Proof Explorer


Theorem 4atlem4a

Description: Lemma for 4at . Frequently used associative law. (Contributed by NM, 9-Jul-2012)

Ref Expression
Hypotheses 4at.l ˙ = K
4at.j ˙ = join K
4at.a A = Atoms K
Assertion 4atlem4a K HL P A Q A R A S A P ˙ Q ˙ R ˙ S = P ˙ Q ˙ R ˙ S

Proof

Step Hyp Ref Expression
1 4at.l ˙ = K
2 4at.j ˙ = join K
3 4at.a A = Atoms K
4 simpl1 K HL P A Q A R A S A K HL
5 4 hllatd K HL P A Q A R A S A K Lat
6 simpl2 K HL P A Q A R A S A P A
7 eqid Base K = Base K
8 7 3 atbase P A P Base K
9 6 8 syl K HL P A Q A R A S A P Base K
10 simpl3 K HL P A Q A R A S A Q A
11 7 3 atbase Q A Q Base K
12 10 11 syl K HL P A Q A R A S A Q Base K
13 simprl K HL P A Q A R A S A R A
14 simprr K HL P A Q A R A S A S A
15 7 2 3 hlatjcl K HL R A S A R ˙ S Base K
16 4 13 14 15 syl3anc K HL P A Q A R A S A R ˙ S Base K
17 7 2 latjass K Lat P Base K Q Base K R ˙ S Base K P ˙ Q ˙ R ˙ S = P ˙ Q ˙ R ˙ S
18 5 9 12 16 17 syl13anc K HL P A Q A R A S A P ˙ Q ˙ R ˙ S = P ˙ Q ˙ R ˙ S
19 2 3 hlatjass K HL Q A R A S A Q ˙ R ˙ S = Q ˙ R ˙ S
20 4 10 13 14 19 syl13anc K HL P A Q A R A S A Q ˙ R ˙ S = Q ˙ R ˙ S
21 20 oveq2d K HL P A Q A R A S A P ˙ Q ˙ R ˙ S = P ˙ Q ˙ R ˙ S
22 18 21 eqtr4d K HL P A Q A R A S A P ˙ Q ˙ R ˙ S = P ˙ Q ˙ R ˙ S