Metamath Proof Explorer


Theorem cdleme48gfv1

Description: TODO: fix comment. (Contributed by NM, 9-Apr-2013)

Ref Expression
Hypotheses cdlemef46g.b B = Base K
cdlemef46g.l ˙ = K
cdlemef46g.j ˙ = join K
cdlemef46g.m ˙ = meet K
cdlemef46g.a A = Atoms K
cdlemef46g.h H = LHyp K
cdlemef46g.u U = P ˙ Q ˙ W
cdlemef46g.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
cdlemefs46g.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
cdlemef46g.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
cdlemef46.v V = Q ˙ P ˙ W
cdlemef46.n N = v ˙ V ˙ P ˙ Q ˙ v ˙ W
cdlemefs46.o O = Q ˙ P ˙ N ˙ u ˙ v ˙ W
cdlemef46.g G = a B if Q P ¬ a ˙ W ι c B | u A ¬ u ˙ W u ˙ a ˙ W = a c = if u ˙ Q ˙ P ι b B | v A ¬ v ˙ W ¬ v ˙ Q ˙ P b = O u / v N ˙ a ˙ W a
Assertion cdleme48gfv1 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W G F X = X

Proof

Step Hyp Ref Expression
1 cdlemef46g.b B = Base K
2 cdlemef46g.l ˙ = K
3 cdlemef46g.j ˙ = join K
4 cdlemef46g.m ˙ = meet K
5 cdlemef46g.a A = Atoms K
6 cdlemef46g.h H = LHyp K
7 cdlemef46g.u U = P ˙ Q ˙ W
8 cdlemef46g.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
9 cdlemefs46g.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
10 cdlemef46g.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 cdlemef46.v V = Q ˙ P ˙ W
12 cdlemef46.n N = v ˙ V ˙ P ˙ Q ˙ v ˙ W
13 cdlemefs46.o O = Q ˙ P ˙ N ˙ u ˙ v ˙ W
14 cdlemef46.g G = a B if Q P ¬ a ˙ W ι c B | u A ¬ u ˙ W u ˙ a ˙ W = a c = if u ˙ Q ˙ P ι b B | v A ¬ v ˙ W ¬ v ˙ Q ˙ P b = O u / v N ˙ a ˙ W a
15 simpl1 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W K HL W H
16 simprr K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W X B ¬ X ˙ W
17 1 2 3 4 5 6 lhpmcvr2 K HL W H X B ¬ X ˙ W e A ¬ e ˙ W e ˙ X ˙ W = X
18 15 16 17 syl2anc K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W e A ¬ e ˙ W e ˙ X ˙ W = X
19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme48d K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W e A ¬ e ˙ W e ˙ X ˙ W = X G F X = X
20 19 3expia K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W e A ¬ e ˙ W e ˙ X ˙ W = X G F X = X
21 20 exp4c K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W e A ¬ e ˙ W e ˙ X ˙ W = X G F X = X
22 21 imp4a K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W e A ¬ e ˙ W e ˙ X ˙ W = X G F X = X
23 22 rexlimdv K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W e A ¬ e ˙ W e ˙ X ˙ W = X G F X = X
24 18 23 mpd K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q X B ¬ X ˙ W G F X = X