Metamath Proof Explorer


Theorem cdlemky

Description: Part of proof of Lemma K of Crawley p. 118. TODO: clean up ( b Y G ) stuff. V represents Y in cdlemk31 . (Contributed by NM, 21-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
cdlemk5b.s S = f T ι i T | i P = P ˙ R f ˙ N P ˙ R f F -1
cdlemk5b.u1 V = d T , e T ι j T | j P = P ˙ R e ˙ S d P ˙ R e d -1
Assertion cdlemky K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G G / g Y = b V G P

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 cdlemk5b.s S = f T ι i T | i P = P ˙ R f ˙ N P ˙ R f F -1
12 cdlemk5b.u1 V = d T , e T ι j T | j P = P ˙ R e ˙ S d P ˙ R e d -1
13 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 b T b I B R b R F R b R G K HL W H
14 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 b T b I B R b R F R b R G R F = R N
15 simp12l K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G F T
16 simp3l K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G b T
17 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 b T b I B R b R F R b R G N T
18 simp3r2 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G R b R F
19 simp12r K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G F I B
20 simp3r1 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G b I B
21 19 20 jca K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G F I B b I B
22 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 b T b I B R b R F R b R G P A ¬ P ˙ W
23 1 2 3 4 5 6 7 8 11 cdlemk30 K HL W H R F = R N F T b T N T R b R F F I B b I B P A ¬ P ˙ W S b P = P ˙ R b ˙ N P ˙ R b F -1
24 13 14 15 16 17 18 21 22 23 syl233anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G S b P = P ˙ R b ˙ N P ˙ R b F -1
25 24 9 eqtr4di K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G S b P = Z
26 25 oveq1d K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G S b P ˙ R G b -1 = Z ˙ R G b -1
27 26 oveq2d K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G P ˙ R G ˙ S b P ˙ R G b -1 = P ˙ R G ˙ Z ˙ R G b -1
28 15 16 17 3jca K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G F T b T N T
29 simp13l K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G G T
30 simp3r3 K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G R b R G
31 18 30 jca K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G R b R F R b R G
32 simp13r K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G G I B
33 19 20 32 3jca K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G F I B b I B G I B
34 1 2 3 4 5 6 7 8 11 12 cdlemk31 K HL W H R F = R N F T b T N T G T R b R F R b R G F I B b I B G I B P A ¬ P ˙ W b V G P = P ˙ R G ˙ S b P ˙ R G b -1
35 13 14 28 29 31 33 22 34 syl223anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G b V G P = P ˙ R G ˙ S b P ˙ R G b -1
36 10 cdlemk41 G T G / g Y = P ˙ R G ˙ Z ˙ R G b -1
37 29 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 b T b I B R b R F R b R G G / g Y = P ˙ R G ˙ Z ˙ R G b -1
38 27 35 37 3eqtr4rd K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G G / g Y = b V G P