Metamath Proof Explorer


Theorem cdlemg18

Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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 cdlemg18 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r P ˙ F G P ˙ 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 simp11 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r K HL W H
9 simp21r K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r G T
10 simp12 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r P A ¬ P ˙ W
11 1 2 3 4 5 6 7 cdlemg18d K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r P ˙ F G P ˙ Q ˙ F G Q A
12 simp23 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r G P P
13 simp1 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W
14 simp21l K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r F T
15 simp22 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r P Q
16 simp31 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r R G ˙ P ˙ Q
17 simp33 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r
18 1 2 3 4 5 6 7 cdlemg17 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r G P ˙ F G P ˙ Q ˙ F G Q = P ˙ F G P ˙ Q ˙ F G Q
19 13 14 9 15 12 16 17 18 syl133anc K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r G P ˙ F G P ˙ Q ˙ F G Q = P ˙ F G P ˙ Q ˙ F G Q
20 1 4 5 6 ltrnatlw K HL W H G T P A ¬ P ˙ W P ˙ F G P ˙ Q ˙ F G Q A G P P G P ˙ F G P ˙ Q ˙ F G Q = P ˙ F G P ˙ Q ˙ F G Q P ˙ F G P ˙ Q ˙ F G Q ˙ W
21 8 9 10 11 12 19 20 syl132anc K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F T G T P Q G P P R G ˙ P ˙ Q F G P ˙ F G Q P ˙ Q ¬ r A ¬ r ˙ W P ˙ r = Q ˙ r P ˙ F G P ˙ Q ˙ F G Q ˙ W