Metamath Proof Explorer


Theorem cdlemk49

Description: Part of proof of Lemma K of Crawley p. 118. Line 5, p. 120. G , I stand for g, h. X represents tau. (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 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

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 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
13 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
14 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
15 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
16 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
17 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
18 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
19 12 13 14 15 16 17 18 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
20 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
21 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
22 12 13 20 15 16 17 21 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
23 6 7 ltrncom K HL W H G / g X T I / g X T G / g X I / g X = I / g X G / g X
24 12 19 22 23 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 = I / g X G / g X
25 24 fveq1d 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 G / g X P
26 simp2 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 P A ¬ P ˙ W R F = R N
27 1 2 3 4 5 6 7 8 9 10 11 cdlemk48 K HL W H F T F I B I T I I B N T P A ¬ P ˙ W R F = R N G T G I B I / g X G / g X P ˙ G / g X P ˙ R I / g X
28 12 13 20 26 14 27 syl311anc 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 G / g X P ˙ G / g X P ˙ R I / g X
29 25 28 eqbrtrd 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