Metamath Proof Explorer


Theorem cdleme50trn1

Description: Part of proof that F is a translation. -. R .<_ ( P .\/ Q ) case. TODO: fix comment. (Contributed by NM, 10-Apr-2013)

Ref Expression
Hypotheses cdlemef50.b B = Base K
cdlemef50.l ˙ = K
cdlemef50.j ˙ = join K
cdlemef50.m ˙ = meet K
cdlemef50.a A = Atoms K
cdlemef50.h H = LHyp K
cdlemef50.u U = P ˙ Q ˙ W
cdlemef50.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
cdlemefs50.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
cdlemef50.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
Assertion cdleme50trn1 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R ˙ F R ˙ W = U

Proof

Step Hyp Ref Expression
1 cdlemef50.b B = Base K
2 cdlemef50.l ˙ = K
3 cdlemef50.j ˙ = join K
4 cdlemef50.m ˙ = meet K
5 cdlemef50.a A = Atoms K
6 cdlemef50.h H = LHyp K
7 cdlemef50.u U = P ˙ Q ˙ W
8 cdlemef50.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
9 cdlemefs50.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
10 cdlemef50.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
11 1 2 3 4 5 6 7 8 10 cdlemefr45e K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q F R = R ˙ U ˙ Q ˙ P ˙ R ˙ W
12 11 oveq2d K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R ˙ F R = R ˙ R ˙ U ˙ Q ˙ P ˙ R ˙ W
13 12 oveq1d K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R ˙ F R ˙ W = R ˙ R ˙ U ˙ Q ˙ P ˙ R ˙ W ˙ W
14 simp11 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q K HL W H
15 simp12l K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q P A
16 simp13l K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q Q A
17 simp2r K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R A ¬ R ˙ W
18 eqid R ˙ U ˙ Q ˙ P ˙ R ˙ W = R ˙ U ˙ Q ˙ P ˙ R ˙ W
19 2 3 4 5 6 7 18 cdleme2 K HL W H P A Q A R A ¬ R ˙ W R ˙ R ˙ U ˙ Q ˙ P ˙ R ˙ W ˙ W = U
20 14 15 16 17 19 syl13anc K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R ˙ R ˙ U ˙ Q ˙ P ˙ R ˙ W ˙ W = U
21 13 20 eqtrd K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W ¬ R ˙ P ˙ Q R ˙ F R ˙ W = U