Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Norm Megill
Construction of a vector space from a Hilbert lattice
erngmul-rN
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Metamath Proof Explorer
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Unicode
Theorem
erngmul-rN
Description:
Ring addition operation.
(Contributed by
NM
, 10-Jun-2013)
(New usage is discouraged.)
Ref
Expression
Hypotheses
erngset.h-r
⊢
H
=
LHyp
⁡
K
erngset.t-r
⊢
T
=
LTrn
⁡
K
⁡
W
erngset.e-r
⊢
E
=
TEndo
⁡
K
⁡
W
erngset.d-r
⊢
D
=
EDRing
R
⁡
K
⁡
W
erng.m-r
⊢
·
˙
=
⋅
D
Assertion
erngmul-rN
⊢
K
∈
X
∧
W
∈
H
∧
U
∈
E
∧
V
∈
E
→
U
·
˙
V
=
V
∘
U
Proof
Step
Hyp
Ref
Expression
1
erngset.h-r
⊢
H
=
LHyp
⁡
K
2
erngset.t-r
⊢
T
=
LTrn
⁡
K
⁡
W
3
erngset.e-r
⊢
E
=
TEndo
⁡
K
⁡
W
4
erngset.d-r
⊢
D
=
EDRing
R
⁡
K
⁡
W
5
erng.m-r
⊢
·
˙
=
⋅
D
6
1
2
3
4
5
erngfmul-rN
⊢
K
∈
X
∧
W
∈
H
→
·
˙
=
s
∈
E
,
t
∈
E
⟼
t
∘
s
7
6
adantr
⊢
K
∈
X
∧
W
∈
H
∧
U
∈
E
∧
V
∈
E
→
·
˙
=
s
∈
E
,
t
∈
E
⟼
t
∘
s
8
7
oveqd
⊢
K
∈
X
∧
W
∈
H
∧
U
∈
E
∧
V
∈
E
→
U
·
˙
V
=
U
s
∈
E
,
t
∈
E
⟼
t
∘
s
V
9
coexg
⊢
V
∈
E
∧
U
∈
E
→
V
∘
U
∈
V
10
9
ancoms
⊢
U
∈
E
∧
V
∈
E
→
V
∘
U
∈
V
11
coeq2
⊢
s
=
U
→
t
∘
s
=
t
∘
U
12
coeq1
⊢
t
=
V
→
t
∘
U
=
V
∘
U
13
eqid
⊢
s
∈
E
,
t
∈
E
⟼
t
∘
s
=
s
∈
E
,
t
∈
E
⟼
t
∘
s
14
11
12
13
ovmpog
⊢
U
∈
E
∧
V
∈
E
∧
V
∘
U
∈
V
→
U
s
∈
E
,
t
∈
E
⟼
t
∘
s
V
=
V
∘
U
15
10
14
mpd3an3
⊢
U
∈
E
∧
V
∈
E
→
U
s
∈
E
,
t
∈
E
⟼
t
∘
s
V
=
V
∘
U
16
15
adantl
⊢
K
∈
X
∧
W
∈
H
∧
U
∈
E
∧
V
∈
E
→
U
s
∈
E
,
t
∈
E
⟼
t
∘
s
V
=
V
∘
U
17
8
16
eqtrd
⊢
K
∈
X
∧
W
∈
H
∧
U
∈
E
∧
V
∈
E
→
U
·
˙
V
=
V
∘
U