Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Norm Megill
Projective geometries based on Hilbert lattices
psubspi
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psubspi2N
Metamath Proof Explorer
Ascii
Unicode
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