Metamath Proof Explorer


Theorem cdlemefs45

Description: Value of f_s(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO: FIX COMMENT. (Contributed by NM, 1-Apr-2013)

Ref Expression
Hypotheses cdlemef45.b B = Base K
cdlemef45.l ˙ = K
cdlemef45.j ˙ = join K
cdlemef45.m ˙ = meet K
cdlemef45.a A = Atoms K
cdlemef45.h H = LHyp K
cdlemef45.u U = P ˙ Q ˙ W
cdlemef45.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
cdlemef45.f F = x B if P Q ¬ x ˙ W ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = if s ˙ P ˙ Q ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E s / t D ˙ x ˙ W x
cdlemefs45.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
Assertion cdlemefs45 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W R ˙ P ˙ Q ¬ S ˙ P ˙ Q F R = R / s S / t E

Proof

Step Hyp Ref Expression
1 cdlemef45.b B = Base K
2 cdlemef45.l ˙ = K
3 cdlemef45.j ˙ = join K
4 cdlemef45.m ˙ = meet K
5 cdlemef45.a A = Atoms K
6 cdlemef45.h H = LHyp K
7 cdlemef45.u U = P ˙ Q ˙ W
8 cdlemef45.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
9 cdlemef45.f F = x B if P Q ¬ x ˙ W ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = if s ˙ P ˙ Q ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E s / t D ˙ x ˙ W x
10 cdlemefs45.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
11 eqid ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = if s ˙ P ˙ Q ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E s / t D ˙ x ˙ W = ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = if s ˙ P ˙ Q ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E s / t D ˙ x ˙ W
12 eqid ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E = ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E
13 1 2 3 4 5 6 7 8 11 9 10 12 cdlemefs44 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W R ˙ P ˙ Q ¬ S ˙ P ˙ Q F R = R / s S / t E