Metamath Proof Explorer


Theorem cdlemg28a

Description: Part of proof of Lemma G of Crawley p. 116. First equality of the equation of line 14 on p. 117. (Contributed by NM, 29-May-2013)

Ref Expression
Hypotheses cdlemg12.l ˙ = K
cdlemg12.j ˙ = join K
cdlemg12.m ˙ = meet K
cdlemg12.a A = Atoms K
cdlemg12.h H = LHyp K
cdlemg12.t T = LTrn K W
cdlemg12b.r R = trL K W
Assertion cdlemg28a K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P P ˙ F G P ˙ W = z ˙ F G z ˙ W

Proof

Step Hyp Ref Expression
1 cdlemg12.l ˙ = K
2 cdlemg12.j ˙ = join K
3 cdlemg12.m ˙ = meet K
4 cdlemg12.a A = Atoms K
5 cdlemg12.h H = LHyp K
6 cdlemg12.t T = LTrn K W
7 cdlemg12b.r R = trL K W
8 simp11 K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P K HL W H
9 simp12 K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P P A ¬ P ˙ W
10 simp21 K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P z A ¬ z ˙ W
11 simp22 K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P F T
12 simp23 K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P G T
13 simp1 K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P K HL W H P A ¬ P ˙ W v A v ˙ W
14 simp21l K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P z A
15 simp31l K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P v R F
16 simp32 K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P z ˙ P ˙ v
17 simp33l K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P F P P
18 1 2 3 4 5 6 7 cdlemg27a K HL W H P A ¬ P ˙ W v A v ˙ W z A F T v R F z ˙ P ˙ v F P P ¬ R F ˙ P ˙ z
19 13 14 11 15 16 17 18 syl123anc K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P ¬ R F ˙ P ˙ z
20 simp31r K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P v R G
21 simp33r K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P G P P
22 1 2 3 4 5 6 7 cdlemg27a K HL W H P A ¬ P ˙ W v A v ˙ W z A G T v R G z ˙ P ˙ v G P P ¬ R G ˙ P ˙ z
23 13 14 12 20 16 21 22 syl123anc K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P ¬ R G ˙ P ˙ z
24 1 2 3 4 5 6 7 cdlemg25zz K HL W H P A ¬ P ˙ W z A ¬ z ˙ W F T G T ¬ R F ˙ P ˙ z ¬ R G ˙ P ˙ z P ˙ F G P ˙ W = z ˙ F G z ˙ W
25 8 9 10 11 12 19 23 24 syl133anc K HL W H P A ¬ P ˙ W v A v ˙ W z A ¬ z ˙ W F T G T v R F v R G z ˙ P ˙ v F P P G P P P ˙ F G P ˙ W = z ˙ F G z ˙ W