Metamath Proof Explorer


Theorem atexchltN

Description: Atom exchange property. Version of hlatexch2 with less-than ordering. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses atexchlt.s < ˙ = < K
atexchlt.j ˙ = join K
atexchlt.a A = Atoms K
Assertion atexchltN K HL P A Q A R A P R P < ˙ Q ˙ R Q < ˙ P ˙ R

Proof

Step Hyp Ref Expression
1 atexchlt.s < ˙ = < K
2 atexchlt.j ˙ = join K
3 atexchlt.a A = Atoms K
4 eqid K = K
5 2 3 4 atexchcvrN K HL P A Q A R A P R P K Q ˙ R Q K P ˙ R
6 1 2 3 4 atltcvr K HL P A Q A R A P < ˙ Q ˙ R P K Q ˙ R
7 6 3adant3 K HL P A Q A R A P R P < ˙ Q ˙ R P K Q ˙ R
8 simpl K HL P A Q A R A K HL
9 simpr2 K HL P A Q A R A Q A
10 simpr1 K HL P A Q A R A P A
11 simpr3 K HL P A Q A R A R A
12 1 2 3 4 atltcvr K HL Q A P A R A Q < ˙ P ˙ R Q K P ˙ R
13 8 9 10 11 12 syl13anc K HL P A Q A R A Q < ˙ P ˙ R Q K P ˙ R
14 13 3adant3 K HL P A Q A R A P R Q < ˙ P ˙ R Q K P ˙ R
15 5 7 14 3imtr4d K HL P A Q A R A P R P < ˙ Q ˙ R Q < ˙ P ˙ R