Metamath Proof Explorer


Theorem psubspi

Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012)

Ref Expression
Hypotheses psubspset.l ˙ = K
psubspset.j ˙ = join K
psubspset.a A = Atoms K
psubspset.s S = PSubSp K
Assertion psubspi K D X S P A q X r X P ˙ q ˙ r P X

Proof

Step Hyp Ref Expression
1 psubspset.l ˙ = K
2 psubspset.j ˙ = join K
3 psubspset.a A = Atoms K
4 psubspset.s S = PSubSp K
5 1 2 3 4 ispsubsp2 K D X S X A p A q X r X p ˙ q ˙ r p X
6 5 simplbda K D X S p A q X r X p ˙ q ˙ r p X
7 6 ex K D X S p A q X r X p ˙ q ˙ r p X
8 breq1 p = P p ˙ q ˙ r P ˙ q ˙ r
9 8 2rexbidv p = P q X r X p ˙ q ˙ r q X r X P ˙ q ˙ r
10 eleq1 p = P p X P X
11 9 10 imbi12d p = P q X r X p ˙ q ˙ r p X q X r X P ˙ q ˙ r P X
12 11 rspccv p A q X r X p ˙ q ˙ r p X P A q X r X P ˙ q ˙ r P X
13 7 12 syl6 K D X S P A q X r X P ˙ q ˙ r P X
14 13 3imp1 K D X S P A q X r X P ˙ q ˙ r P X