Metamath Proof Explorer


Theorem cdlemkfid3N

Description: TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013) (New usage is discouraged.)

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
Assertion cdlemkfid3N K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W G / g Y = 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 simp22 K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W G T
12 10 cdlemk41 G T G / g Y = P ˙ R G ˙ Z ˙ R G b -1
13 11 12 syl K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W G / g Y = P ˙ R G ˙ Z ˙ R G b -1
14 simp1 K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W K HL W H F = N
15 simp21l K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W F T
16 simp21r K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W F I B
17 simp23l K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W b T
18 simp31 K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W R b R F
19 simp33 K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W P A ¬ P ˙ W
20 1 2 3 4 5 6 7 8 9 cdlemkfid2N K HL W H F = N F T F I B b T R b R F P A ¬ P ˙ W Z = b P
21 14 15 16 17 18 19 20 syl132anc K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W Z = b P
22 21 oveq1d K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W Z ˙ R G b -1 = b P ˙ R G b -1
23 22 oveq2d K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W P ˙ R G ˙ Z ˙ R G b -1 = P ˙ R G ˙ b P ˙ R G b -1
24 simp1l K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W K HL W H
25 simp23r K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W b I B
26 simp32 K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W R b R G
27 26 necomd K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W R G R b
28 1 2 3 4 5 6 7 8 cdlemkfid1N K HL W H b T b I B G T R G R b P A ¬ P ˙ W P ˙ R G ˙ b P ˙ R G b -1 = G P
29 24 17 25 11 27 19 28 syl132anc K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W P ˙ R G ˙ b P ˙ R G b -1 = G P
30 13 23 29 3eqtrd K HL W H F = N F T F I B G T b T b I B R b R F R b R G P A ¬ P ˙ W G / g Y = G P