Metamath Proof Explorer


Theorem cdlemk50

Description: Part of proof of Lemma K of Crawley p. 118. Line 6, p. 120. G , I stand for g, h. X represents tau. TODO: Combine into cdlemk52 ? (Contributed by NM, 23-Jul-2013)

Ref Expression
Hypotheses cdlemk5.b B = Base K
cdlemk5.l ˙ = K
cdlemk5.j ˙ = join K
cdlemk5.m ˙ = meet K
cdlemk5.a A = Atoms K
cdlemk5.h H = LHyp K
cdlemk5.t T = LTrn K W
cdlemk5.r R = trL K W
cdlemk5.z Z = P ˙ R b ˙ N P ˙ R b F -1
cdlemk5.y Y = P ˙ R g ˙ Z ˙ R g b -1
cdlemk5.x X = ι z T | b T b I B R b R F R b R g z P = Y
Assertion cdlemk50 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X I / g X P ˙ G / g X P ˙ R I / g X ˙ I / g X P ˙ R G / g X

Proof

Step Hyp Ref Expression
1 cdlemk5.b B = Base K
2 cdlemk5.l ˙ = K
3 cdlemk5.j ˙ = join K
4 cdlemk5.m ˙ = meet K
5 cdlemk5.a A = Atoms K
6 cdlemk5.h H = LHyp K
7 cdlemk5.t T = LTrn K W
8 cdlemk5.r R = trL K W
9 cdlemk5.z Z = P ˙ R b ˙ N P ˙ R b F -1
10 cdlemk5.y Y = P ˙ R g ˙ Z ˙ R g b -1
11 cdlemk5.x X = ι z T | b T b I B R b R F R b R g z P = Y
12 1 2 3 4 5 6 7 8 9 10 11 cdlemk49 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X I / g X P ˙ G / g X P ˙ R I / g X
13 1 2 3 4 5 6 7 8 9 10 11 cdlemk48 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X I / g X P ˙ I / g X P ˙ R G / g X
14 simp11l K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B K HL
15 14 hllatd K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B K Lat
16 simp11 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B K HL W H
17 simp12 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B F T F I B
18 simp13 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G T G I B
19 simp21 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B N T
20 simp22 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B P A ¬ P ˙ W
21 simp23 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B R F = R N
22 1 2 3 4 5 6 7 8 9 10 11 cdlemk35s K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N G / g X T
23 16 17 18 19 20 21 22 syl132anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X T
24 simp3 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B I T I I B
25 1 2 3 4 5 6 7 8 9 10 11 cdlemk35s K HL W H F T F I B I T I I B N T P A ¬ P ˙ W R F = R N I / g X T
26 16 17 24 19 20 21 25 syl132anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B I / g X T
27 6 7 ltrnco K HL W H G / g X T I / g X T G / g X I / g X T
28 16 23 26 27 syl3anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X I / g X T
29 simp22l K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B P A
30 2 5 6 7 ltrnat K HL W H G / g X I / g X T P A G / g X I / g X P A
31 16 28 29 30 syl3anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X I / g X P A
32 1 5 atbase G / g X I / g X P A G / g X I / g X P B
33 31 32 syl K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X I / g X P B
34 2 5 6 7 ltrnat K HL W H G / g X T P A G / g X P A
35 16 23 29 34 syl3anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X P A
36 1 5 atbase G / g X P A G / g X P B
37 35 36 syl K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X P B
38 1 6 7 8 trlcl K HL W H I / g X T R I / g X B
39 16 26 38 syl2anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B R I / g X B
40 1 3 latjcl K Lat G / g X P B R I / g X B G / g X P ˙ R I / g X B
41 15 37 39 40 syl3anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X P ˙ R I / g X B
42 2 5 6 7 ltrnat K HL W H I / g X T P A I / g X P A
43 16 26 29 42 syl3anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B I / g X P A
44 1 5 atbase I / g X P A I / g X P B
45 43 44 syl K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B I / g X P B
46 1 6 7 8 trlcl K HL W H G / g X T R G / g X B
47 16 23 46 syl2anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B R G / g X B
48 1 3 latjcl K Lat I / g X P B R G / g X B I / g X P ˙ R G / g X B
49 15 45 47 48 syl3anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B I / g X P ˙ R G / g X B
50 1 2 4 latlem12 K Lat G / g X I / g X P B G / g X P ˙ R I / g X B I / g X P ˙ R G / g X B G / g X I / g X P ˙ G / g X P ˙ R I / g X G / g X I / g X P ˙ I / g X P ˙ R G / g X G / g X I / g X P ˙ G / g X P ˙ R I / g X ˙ I / g X P ˙ R G / g X
51 15 33 41 49 50 syl13anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X I / g X P ˙ G / g X P ˙ R I / g X G / g X I / g X P ˙ I / g X P ˙ R G / g X G / g X I / g X P ˙ G / g X P ˙ R I / g X ˙ I / g X P ˙ R G / g X
52 12 13 51 mpbi2and K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N I T I I B G / g X I / g X P ˙ G / g X P ˙ R I / g X ˙ I / g X P ˙ R G / g X