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Metamath Proof Explorer

Theorem chjcom

Description: Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chjcom ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 chsh ⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2 chsh ⊢ ( 𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ )
3 shjcom ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )