Theorem chlej2

Description: Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chlej2	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		chsh	⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		chsh	⊢ ( 𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ )
3		chsh	⊢ ( 𝐶 ∈ Cℋ → 𝐶 ∈ Sℋ )
4		shlej2	⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )
5	1 2 3 4	syl3anl	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )