Metamath Proof Explorer


Theorem cdlemk21-2N

Description: Part of proof of Lemma K of Crawley p. 118. Lines 26-27, p. 119 for i=0 and j=2. (Contributed by NM, 5-Jul-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemk2.b B = Base K
cdlemk2.l ˙ = K
cdlemk2.j ˙ = join K
cdlemk2.m ˙ = meet K
cdlemk2.a A = Atoms K
cdlemk2.h H = LHyp K
cdlemk2.t T = LTrn K W
cdlemk2.r R = trL K W
cdlemk2.s S = f T ι i T | i P = P ˙ R f ˙ N P ˙ R f F -1
cdlemk2.q Q = S C
cdlemk2.v V = d T ι k T | k P = P ˙ R d ˙ Q P ˙ R d C -1
Assertion cdlemk21-2N K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W S G P = V G P

Proof

Step Hyp Ref Expression
1 cdlemk2.b B = Base K
2 cdlemk2.l ˙ = K
3 cdlemk2.j ˙ = join K
4 cdlemk2.m ˙ = meet K
5 cdlemk2.a A = Atoms K
6 cdlemk2.h H = LHyp K
7 cdlemk2.t T = LTrn K W
8 cdlemk2.r R = trL K W
9 cdlemk2.s S = f T ι i T | i P = P ˙ R f ˙ N P ˙ R f F -1
10 cdlemk2.q Q = S C
11 cdlemk2.v V = d T ι k T | k P = P ˙ R d ˙ Q P ˙ R d C -1
12 simp11 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W K HL
13 simp12 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W W H
14 12 13 jca K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W K HL W H
15 simp2l1 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W F T
16 simp2l2 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W C T
17 simp2l3 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W N T
18 simp2rl K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W G T
19 17 18 jca K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W N T G T
20 simp33 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W P A ¬ P ˙ W
21 simp13 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W R F = R N
22 simp322 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W F I B
23 simp323 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W C I B
24 simp2rr K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W G I B
25 22 23 24 3jca K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W F I B C I B G I B
26 simp31l K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W R C R F
27 simp31r K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W R G R C
28 simp321 K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W R G R F
29 26 27 28 3jca K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W R C R F R G R C R G R F
30 1 2 3 4 5 6 7 8 9 10 11 cdlemk21N K HL W H F T C T N T G T P A ¬ P ˙ W R F = R N F I B C I B G I B R C R F R G R C R G R F S G P = V G P
31 14 15 16 19 20 21 25 29 30 syl332anc K HL W H R F = R N F T C T N T G T G I B R C R F R G R C R G R F F I B C I B P A ¬ P ˙ W S G P = V G P