Metamath Proof Explorer


Theorem cdleme42mN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT . f preserves join: f(r \/ s) = f(r) \/ s, p. 115 10th line from bottom. (Contributed by NM, 20-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme41.b B = Base K
cdleme41.l ˙ = K
cdleme41.j ˙ = join K
cdleme41.m ˙ = meet K
cdleme41.a A = Atoms K
cdleme41.h H = LHyp K
cdleme41.u U = P ˙ Q ˙ W
cdleme41.d D = s ˙ U ˙ Q ˙ P ˙ s ˙ W
cdleme41.e E = t ˙ U ˙ Q ˙ P ˙ t ˙ W
cdleme41.g G = P ˙ Q ˙ E ˙ s ˙ t ˙ W
cdleme41.i I = ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = G
cdleme41.n N = if s ˙ P ˙ Q I D
cdleme41.o O = ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = N ˙ x ˙ W
cdleme41.f F = x B if P Q ¬ x ˙ W O x
Assertion cdleme42mN K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W F R ˙ S = F R ˙ F S

Proof

Step Hyp Ref Expression
1 cdleme41.b B = Base K
2 cdleme41.l ˙ = K
3 cdleme41.j ˙ = join K
4 cdleme41.m ˙ = meet K
5 cdleme41.a A = Atoms K
6 cdleme41.h H = LHyp K
7 cdleme41.u U = P ˙ Q ˙ W
8 cdleme41.d D = s ˙ U ˙ Q ˙ P ˙ s ˙ W
9 cdleme41.e E = t ˙ U ˙ Q ˙ P ˙ t ˙ W
10 cdleme41.g G = P ˙ Q ˙ E ˙ s ˙ t ˙ W
11 cdleme41.i I = ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = G
12 cdleme41.n N = if s ˙ P ˙ Q I D
13 cdleme41.o O = ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = N ˙ x ˙ W
14 cdleme41.f F = x B if P Q ¬ x ˙ W O x
15 simpl K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W
16 simpr2 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W R A ¬ R ˙ W
17 simpr3 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W S A ¬ S ˙ W
18 simpr1 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W P Q
19 eqid R ˙ S ˙ W = R ˙ S ˙ W
20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 19 cdleme42g K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W R A ¬ R ˙ W S A ¬ S ˙ W P Q F R ˙ S = F R ˙ R ˙ S ˙ W
21 15 16 17 18 20 syl121anc K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W F R ˙ S = F R ˙ R ˙ S ˙ W
22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 19 cdleme42ke K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W F R ˙ F S = F R ˙ R ˙ S ˙ W
23 21 22 eqtr4d K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W S A ¬ S ˙ W F R ˙ S = F R ˙ F S