Metamath Proof Explorer


Theorem dihjatb

Description: Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014)

Ref Expression
Hypotheses dihjatb.l ˙ = K
dihjatb.h H = LHyp K
dihjatb.j ˙ = join K
dihjatb.a A = Atoms K
dihjatb.u U = DVecH K W
dihjatb.s ˙ = LSSum U
dihjatb.i I = DIsoH K W
dihjatb.k φ K HL W H
dihjatb.p φ P A P ˙ W
dihjatb.q φ Q A Q ˙ W
Assertion dihjatb φ I P ˙ Q = I P ˙ I Q

Proof

Step Hyp Ref Expression
1 dihjatb.l ˙ = K
2 dihjatb.h H = LHyp K
3 dihjatb.j ˙ = join K
4 dihjatb.a A = Atoms K
5 dihjatb.u U = DVecH K W
6 dihjatb.s ˙ = LSSum U
7 dihjatb.i I = DIsoH K W
8 dihjatb.k φ K HL W H
9 dihjatb.p φ P A P ˙ W
10 dihjatb.q φ Q A Q ˙ W
11 1 3 4 2 5 6 7 8 9 10 dih2dimb φ I P ˙ Q I P ˙ I Q
12 eqid Base K = Base K
13 9 simpld φ P A
14 12 4 atbase P A P Base K
15 13 14 syl φ P Base K
16 10 simpld φ Q A
17 12 4 atbase Q A Q Base K
18 16 17 syl φ Q Base K
19 12 2 3 5 6 7 8 15 18 dihsumssj φ I P ˙ I Q I P ˙ Q
20 11 19 eqssd φ I P ˙ Q = I P ˙ I Q