Theorem chlej12i

Description: Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	⊢ 𝐴 ∈ Cℋ
	chjcl.2	⊢ 𝐵 ∈ Cℋ
	chlub.1	⊢ 𝐶 ∈ Cℋ
	chlej12.4	⊢ 𝐷 ∈ Cℋ
Assertion	chlej12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3		chlub.1	⊢ 𝐶 ∈ Cℋ
4		chlej12.4	⊢ 𝐷 ∈ Cℋ
5	1 2 3	chlej1i	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )
6	3 4 2	chlej2i	⊢ ( 𝐶 ⊆ 𝐷 → ( 𝐵 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐷 ) )
7	5 6	sylan9ss	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐷 ) )