Metamath Proof Explorer


Theorem cdlemg11b

Description: TODO: FIX COMMENT. (Contributed by NM, 5-May-2013)

Ref Expression
Hypotheses cdlemg8.l ˙ = K
cdlemg8.j ˙ = join K
cdlemg8.m ˙ = meet K
cdlemg8.a A = Atoms K
cdlemg8.h H = LHyp K
cdlemg8.t T = LTrn K W
cdlemg10.r R = trL K W
Assertion cdlemg11b K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q G P ˙ G Q

Proof

Step Hyp Ref Expression
1 cdlemg8.l ˙ = K
2 cdlemg8.j ˙ = join K
3 cdlemg8.m ˙ = meet K
4 cdlemg8.a A = Atoms K
5 cdlemg8.h H = LHyp K
6 cdlemg8.t T = LTrn K W
7 cdlemg10.r R = trL K W
8 simp33 K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q ¬ R G ˙ P ˙ Q
9 simpl1 K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q K HL W H
10 simpl31 K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q G T
11 simpl2l K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P A ¬ P ˙ W
12 1 2 3 4 5 6 7 trlval2 K HL W H G T P A ¬ P ˙ W R G = P ˙ G P ˙ W
13 9 10 11 12 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q R G = P ˙ G P ˙ W
14 eqid Base K = Base K
15 simpl1l K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q K HL
16 15 hllatd K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q K Lat
17 simp2ll K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P A
18 17 adantr K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P A
19 14 4 atbase P A P Base K
20 18 19 syl K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P Base K
21 14 5 6 ltrncl K HL W H G T P Base K G P Base K
22 9 10 20 21 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q G P Base K
23 14 2 latjcl K Lat P Base K G P Base K P ˙ G P Base K
24 16 20 22 23 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ G P Base K
25 simpl1r K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q W H
26 14 5 lhpbase W H W Base K
27 25 26 syl K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q W Base K
28 14 3 latmcl K Lat P ˙ G P Base K W Base K P ˙ G P ˙ W Base K
29 16 24 27 28 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ G P ˙ W Base K
30 simpl2r K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q Q A
31 14 4 atbase Q A Q Base K
32 30 31 syl K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q Q Base K
33 14 2 latjcl K Lat P Base K Q Base K P ˙ Q Base K
34 16 20 32 33 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ Q Base K
35 14 1 3 latmle1 K Lat P ˙ G P Base K W Base K P ˙ G P ˙ W ˙ P ˙ G P
36 16 24 27 35 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ G P ˙ W ˙ P ˙ G P
37 14 1 2 latlej1 K Lat P Base K Q Base K P ˙ P ˙ Q
38 16 20 32 37 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ P ˙ Q
39 14 5 6 ltrncl K HL W H G T Q Base K G Q Base K
40 9 10 32 39 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q G Q Base K
41 14 1 2 latlej1 K Lat G P Base K G Q Base K G P ˙ G P ˙ G Q
42 16 22 40 41 syl3anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q G P ˙ G P ˙ G Q
43 simpr K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ Q = G P ˙ G Q
44 42 43 breqtrrd K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q G P ˙ P ˙ Q
45 14 1 2 latjle12 K Lat P Base K G P Base K P ˙ Q Base K P ˙ P ˙ Q G P ˙ P ˙ Q P ˙ G P ˙ P ˙ Q
46 16 20 22 34 45 syl13anc K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ P ˙ Q G P ˙ P ˙ Q P ˙ G P ˙ P ˙ Q
47 38 44 46 mpbi2and K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ G P ˙ P ˙ Q
48 14 1 16 29 24 34 36 47 lattrd K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q P ˙ G P ˙ W ˙ P ˙ Q
49 13 48 eqbrtrd K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q R G ˙ P ˙ Q
50 49 ex K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q = G P ˙ G Q R G ˙ P ˙ Q
51 50 necon3bd K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q ¬ R G ˙ P ˙ Q P ˙ Q G P ˙ G Q
52 8 51 mpd K HL W H P A ¬ P ˙ W Q A G T P Q ¬ R G ˙ P ˙ Q P ˙ Q G P ˙ G Q