Metamath Proof Explorer


Theorem cdlemg28

Description: Part of proof of Lemma G of Crawley p. 116. Chain the equalities of line 14 on p. 117. TODO: rearrange hypotheses in the order of cdlemg29 (and maybe leading up to this too)? (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
cdlemg31.n N = P ˙ v ˙ Q ˙ R F
cdlemg33.o O = P ˙ v ˙ Q ˙ R G
Assertion cdlemg28 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P P ˙ F G P ˙ W = Q ˙ F G Q ˙ 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 cdlemg31.n N = P ˙ v ˙ Q ˙ R F
9 cdlemg33.o O = P ˙ v ˙ Q ˙ R G
10 simp11 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P K HL W H
11 simp12 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P P A ¬ P ˙ W
12 simp21 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P v A v ˙ W
13 simp22 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P z A ¬ z ˙ W
14 simp23l K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P F T
15 simp23r K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P G T
16 simp32 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P v R F v R G
17 simp313 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P z ˙ P ˙ v
18 simp33 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P F P P G P P
19 1 2 3 4 5 6 7 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
20 10 11 12 13 14 15 16 17 18 19 syl333anc K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P P ˙ F G P ˙ W = z ˙ F G z ˙ W
21 1 2 3 4 5 6 7 8 9 cdlemg28b K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P Q ˙ F G Q ˙ W = z ˙ F G z ˙ W
22 20 21 eqtr4d K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W v A v ˙ W z A ¬ z ˙ W F T G T z N z O z ˙ P ˙ v v R F v R G F P P G P P P ˙ F G P ˙ W = Q ˙ F G Q ˙ W