Metamath Proof Explorer
Description: Closure of the zero functional in the set of functionals with closed
kernels. (Contributed by NM, 15-Mar-2015)
|
|
Ref |
Expression |
|
Hypotheses |
lcdv0cl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
|
|
lcdv0cl.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
|
|
lcdv0cl.v |
⊢ 𝑉 = ( Base ‘ 𝐶 ) |
|
|
lcdv0cl.o |
⊢ 𝑂 = ( 0g ‘ 𝐶 ) |
|
|
lcdv0cl.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
|
Assertion |
lcd0vcl |
⊢ ( 𝜑 → 𝑂 ∈ 𝑉 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
lcdv0cl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lcdv0cl.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lcdv0cl.v |
⊢ 𝑉 = ( Base ‘ 𝐶 ) |
4 |
|
lcdv0cl.o |
⊢ 𝑂 = ( 0g ‘ 𝐶 ) |
5 |
|
lcdv0cl.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
1 2 5
|
lcdlmod |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
7 |
3 4
|
lmod0vcl |
⊢ ( 𝐶 ∈ LMod → 𝑂 ∈ 𝑉 ) |
8 |
6 7
|
syl |
⊢ ( 𝜑 → 𝑂 ∈ 𝑉 ) |