Description: Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | chjidm | ⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ 𝐴 ) = 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inidm | ⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 | |
2 | 1 | oveq2i | ⊢ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐴 ) ) = ( 𝐴 ∨ℋ 𝐴 ) |
3 | chabs1 | ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐴 ) ) = 𝐴 ) | |
4 | 3 | anidms | ⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐴 ) ) = 𝐴 ) |
5 | 2 4 | eqtr3id | ⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ 𝐴 ) = 𝐴 ) |