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Mathbox for Norm Megill
Construction of a vector space from a Hilbert lattice
cdleme9b
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cdleme9
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Unicode
Theorem
cdleme9b
Description:
Utility lemma for Lemma E in
Crawley
p. 113.
(Contributed by
NM
, 9-Oct-2012)
Ref
Expression
Hypotheses
cdleme9b.b
⊢
B
=
Base
K
cdleme9b.j
⊢
∨
˙
=
join
⁡
K
cdleme9b.m
⊢
∧
˙
=
meet
⁡
K
cdleme9b.a
⊢
A
=
Atoms
⁡
K
cdleme9b.h
⊢
H
=
LHyp
⁡
K
cdleme9b.c
⊢
C
=
P
∨
˙
S
∧
˙
W
Assertion
cdleme9b
⊢
K
∈
HL
∧
P
∈
A
∧
S
∈
A
∧
W
∈
H
→
C
∈
B
Proof
Step
Hyp
Ref
Expression
1
cdleme9b.b
⊢
B
=
Base
K
2
cdleme9b.j
⊢
∨
˙
=
join
⁡
K
3
cdleme9b.m
⊢
∧
˙
=
meet
⁡
K
4
cdleme9b.a
⊢
A
=
Atoms
⁡
K
5
cdleme9b.h
⊢
H
=
LHyp
⁡
K
6
cdleme9b.c
⊢
C
=
P
∨
˙
S
∧
˙
W
7
hllat
⊢
K
∈
HL
→
K
∈
Lat
8
7
adantr
⊢
K
∈
HL
∧
P
∈
A
∧
S
∈
A
∧
W
∈
H
→
K
∈
Lat
9
1
2
4
hlatjcl
⊢
K
∈
HL
∧
P
∈
A
∧
S
∈
A
→
P
∨
˙
S
∈
B
10
9
3adant3r3
⊢
K
∈
HL
∧
P
∈
A
∧
S
∈
A
∧
W
∈
H
→
P
∨
˙
S
∈
B
11
simpr3
⊢
K
∈
HL
∧
P
∈
A
∧
S
∈
A
∧
W
∈
H
→
W
∈
H
12
1
5
lhpbase
⊢
W
∈
H
→
W
∈
B
13
11
12
syl
⊢
K
∈
HL
∧
P
∈
A
∧
S
∈
A
∧
W
∈
H
→
W
∈
B
14
1
3
latmcl
⊢
K
∈
Lat
∧
P
∨
˙
S
∈
B
∧
W
∈
B
→
P
∨
˙
S
∧
˙
W
∈
B
15
8
10
13
14
syl3anc
⊢
K
∈
HL
∧
P
∈
A
∧
S
∈
A
∧
W
∈
H
→
P
∨
˙
S
∧
˙
W
∈
B
16
6
15
eqeltrid
⊢
K
∈
HL
∧
P
∈
A
∧
S
∈
A
∧
W
∈
H
→
C
∈
B