Metamath Proof Explorer


Theorem cdlemk7u-2N

Description: Part of proof of Lemma K of Crawley p. 118. Line 5, p. 119 for the sigma_2 case. (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
cdlemk2.z Z = G P ˙ X P ˙ R G C -1 ˙ R X C -1
Assertion cdlemk7u-2N K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W V G P ˙ V X P ˙ Z

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 cdlemk2.z Z = G P ˙ X P ˙ R G C -1 ˙ R X C -1
13 simp11 K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W K HL
14 simp12 K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W W H
15 13 14 jca K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W K HL W H
16 simp211 K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W F T
17 simp212 K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W C T
18 simp213 K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W N T
19 simp22l K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W G T
20 simp23l K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W X T
21 18 19 20 3jca K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W N T G T X T
22 simp33 K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W P A ¬ P ˙ W
23 simp13 K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W R F = R N
24 simp32l K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W F I B
25 simp32r K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W C I B
26 simp22r K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W G I B
27 24 25 26 3jca K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W F I B C I B G I B
28 simp23r K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W X I B
29 simp31 K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W R C R F R G R C R X R C
30 1 2 3 4 5 6 7 8 9 10 11 12 cdlemk7u K HL W H F T C T N T G T X T P A ¬ P ˙ W R F = R N F I B C I B G I B X I B R C R F R G R C R X R C V G P ˙ V X P ˙ Z
31 15 16 17 21 22 23 27 28 29 30 syl333anc K HL W H R F = R N F T C T N T G T G I B X T X I B R C R F R G R C R X R C F I B C I B P A ¬ P ˙ W V G P ˙ V X P ˙ Z