Metamath Proof Explorer
Description: Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
shincl.1 |
⊢ 𝐴 ∈ Sℋ |
|
|
shincl.2 |
⊢ 𝐵 ∈ Sℋ |
|
Assertion |
shub1i |
⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
shincl.1 |
⊢ 𝐴 ∈ Sℋ |
2 |
|
shincl.2 |
⊢ 𝐵 ∈ Sℋ |
3 |
1 2
|
shsub1i |
⊢ 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 ) |
4 |
1 2
|
shsleji |
⊢ ( 𝐴 +ℋ 𝐵 ) ⊆ ( 𝐴 ∨ℋ 𝐵 ) |
5 |
3 4
|
sstri |
⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) |