Metamath Proof Explorer


Theorem cdleme50trn123

Description: Part of proof that F is a translation. Combine all cases. 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 cdleme50trn123 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W R A ¬ R ˙ W 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 9 10 cdleme50trn3 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P = Q R A ¬ R ˙ W R ˙ F R ˙ W = U
12 11 anass1rs K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W R A ¬ R ˙ W P = Q R ˙ F R ˙ W = U
13 1 2 3 4 5 6 7 8 9 10 cdleme50trn12 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W R ˙ F R ˙ W = U
14 13 anass1rs K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W R A ¬ R ˙ W P Q R ˙ F R ˙ W = U
15 12 14 pm2.61dane K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W R A ¬ R ˙ W R ˙ F R ˙ W = U