Metamath Proof Explorer


Theorem cdlemefr32snb

Description: Show closure of [_ R / s ]_ N . (Contributed by NM, 28-Mar-2013)

Ref Expression
Hypotheses cdlemefr27.b B = Base K
cdlemefr27.l ˙ = K
cdlemefr27.j ˙ = join K
cdlemefr27.m ˙ = meet K
cdlemefr27.a A = Atoms K
cdlemefr27.h H = LHyp K
cdlemefr27.u U = P ˙ Q ˙ W
cdlemefr27.c C = s ˙ U ˙ Q ˙ P ˙ s ˙ W
cdlemefr27.n N = if s ˙ P ˙ Q I C
Assertion cdlemefr32snb K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R / s N B

Proof

Step Hyp Ref Expression
1 cdlemefr27.b B = Base K
2 cdlemefr27.l ˙ = K
3 cdlemefr27.j ˙ = join K
4 cdlemefr27.m ˙ = meet K
5 cdlemefr27.a A = Atoms K
6 cdlemefr27.h H = LHyp K
7 cdlemefr27.u U = P ˙ Q ˙ W
8 cdlemefr27.c C = s ˙ U ˙ Q ˙ P ˙ s ˙ W
9 cdlemefr27.n N = if s ˙ P ˙ Q I C
10 1 2 3 4 5 6 7 8 9 cdlemefr32sn2aw K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R / s N A ¬ R / s N ˙ W
11 10 simpld K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R / s N A
12 1 5 atbase R / s N A R / s N B
13 11 12 syl K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R / s N B