Metamath Proof Explorer


Theorem cdlemk18-2N

Description: Part of proof of Lemma K of Crawley p. 118. Line 22 on p. 119. N , V , Q , C are k, sigma_2 (p), k_2, f_2. (Contributed by NM, 2-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 cdlemk18-2N K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W N P = V F 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 R C 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 R C 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 R C R F F I B C I B P A ¬ P ˙ W K HL W H
15 simp21 K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W F T
16 simp22 K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W C T
17 simp23 K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W N T
18 simp33 K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W P A ¬ P ˙ W
19 simp13 K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W R F = R N
20 simp32l K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W F I B
21 simp32r K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W C I B
22 simp31 K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W R C R F
23 1 2 3 4 5 6 7 8 9 10 11 cdlemk18 K HL W H F T C T N T P A ¬ P ˙ W R F = R N F I B C I B R C R F N P = V F P
24 14 15 16 17 18 19 20 21 22 23 syl333anc K HL W H R F = R N F T C T N T R C R F F I B C I B P A ¬ P ˙ W N P = V F P