Metamath Proof Explorer


Theorem dihjatcclem3

Description: Lemma for dihjatcc . (Contributed by NM, 28-Sep-2014)

Ref Expression
Hypotheses dihjatcclem.b B = Base K
dihjatcclem.l ˙ = K
dihjatcclem.h H = LHyp K
dihjatcclem.j ˙ = join K
dihjatcclem.m ˙ = meet K
dihjatcclem.a A = Atoms K
dihjatcclem.u U = DVecH K W
dihjatcclem.s ˙ = LSSum U
dihjatcclem.i I = DIsoH K W
dihjatcclem.v V = P ˙ Q ˙ W
dihjatcclem.k φ K HL W H
dihjatcclem.p φ P A ¬ P ˙ W
dihjatcclem.q φ Q A ¬ Q ˙ W
dihjatcc.w C = oc K W
dihjatcc.t T = LTrn K W
dihjatcc.r R = trL K W
dihjatcc.e E = TEndo K W
dihjatcc.g G = ι d T | d C = P
dihjatcc.dd D = ι d T | d C = Q
Assertion dihjatcclem3 φ R G D -1 = V

Proof

Step Hyp Ref Expression
1 dihjatcclem.b B = Base K
2 dihjatcclem.l ˙ = K
3 dihjatcclem.h H = LHyp K
4 dihjatcclem.j ˙ = join K
5 dihjatcclem.m ˙ = meet K
6 dihjatcclem.a A = Atoms K
7 dihjatcclem.u U = DVecH K W
8 dihjatcclem.s ˙ = LSSum U
9 dihjatcclem.i I = DIsoH K W
10 dihjatcclem.v V = P ˙ Q ˙ W
11 dihjatcclem.k φ K HL W H
12 dihjatcclem.p φ P A ¬ P ˙ W
13 dihjatcclem.q φ Q A ¬ Q ˙ W
14 dihjatcc.w C = oc K W
15 dihjatcc.t T = LTrn K W
16 dihjatcc.r R = trL K W
17 dihjatcc.e E = TEndo K W
18 dihjatcc.g G = ι d T | d C = P
19 dihjatcc.dd D = ι d T | d C = Q
20 2 6 3 14 lhpocnel2 K HL W H C A ¬ C ˙ W
21 11 20 syl φ C A ¬ C ˙ W
22 2 6 3 15 18 ltrniotacl K HL W H C A ¬ C ˙ W P A ¬ P ˙ W G T
23 11 21 12 22 syl3anc φ G T
24 2 6 3 15 19 ltrniotacl K HL W H C A ¬ C ˙ W Q A ¬ Q ˙ W D T
25 11 21 13 24 syl3anc φ D T
26 3 15 ltrncnv K HL W H D T D -1 T
27 11 25 26 syl2anc φ D -1 T
28 3 15 ltrnco K HL W H G T D -1 T G D -1 T
29 11 23 27 28 syl3anc φ G D -1 T
30 2 4 5 6 3 15 16 trlval2 K HL W H G D -1 T Q A ¬ Q ˙ W R G D -1 = Q ˙ G D -1 Q ˙ W
31 11 29 13 30 syl3anc φ R G D -1 = Q ˙ G D -1 Q ˙ W
32 13 simpld φ Q A
33 2 6 3 15 ltrncoval K HL W H G T D -1 T Q A G D -1 Q = G D -1 Q
34 11 23 27 32 33 syl121anc φ G D -1 Q = G D -1 Q
35 2 6 3 15 19 ltrniotacnvval K HL W H C A ¬ C ˙ W Q A ¬ Q ˙ W D -1 Q = C
36 11 21 13 35 syl3anc φ D -1 Q = C
37 36 fveq2d φ G D -1 Q = G C
38 2 6 3 15 18 ltrniotaval K HL W H C A ¬ C ˙ W P A ¬ P ˙ W G C = P
39 11 21 12 38 syl3anc φ G C = P
40 37 39 eqtrd φ G D -1 Q = P
41 34 40 eqtrd φ G D -1 Q = P
42 41 oveq2d φ Q ˙ G D -1 Q = Q ˙ P
43 11 simpld φ K HL
44 12 simpld φ P A
45 4 6 hlatjcom K HL P A Q A P ˙ Q = Q ˙ P
46 43 44 32 45 syl3anc φ P ˙ Q = Q ˙ P
47 42 46 eqtr4d φ Q ˙ G D -1 Q = P ˙ Q
48 47 oveq1d φ Q ˙ G D -1 Q ˙ W = P ˙ Q ˙ W
49 48 10 eqtr4di φ Q ˙ G D -1 Q ˙ W = V
50 31 49 eqtrd φ R G D -1 = V