Theorem lejdii

Description: An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ledi.1	⊢ 𝐴 ∈ Cℋ
	ledi.2	⊢ 𝐵 ∈ Cℋ
	ledi.3	⊢ 𝐶 ∈ Cℋ
Assertion	lejdii	⊢ ( 𝐴 ∨ℋ ( 𝐵 ∩ 𝐶 ) ) ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ledi.1	⊢ 𝐴 ∈ Cℋ
2		ledi.2	⊢ 𝐵 ∈ Cℋ
3		ledi.3	⊢ 𝐶 ∈ Cℋ
4	1 2	chub1i	⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 )
5	1 3	chub1i	⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐶 )
6	4 5	ssini	⊢ 𝐴 ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) )
7		inss1	⊢ ( 𝐵 ∩ 𝐶 ) ⊆ 𝐵
8	2 1	chub2i	⊢ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 )
9	7 8	sstri	⊢ ( 𝐵 ∩ 𝐶 ) ⊆ ( 𝐴 ∨ℋ 𝐵 )
10		inss2	⊢ ( 𝐵 ∩ 𝐶 ) ⊆ 𝐶
11	3 1	chub2i	⊢ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐶 )
12	10 11	sstri	⊢ ( 𝐵 ∩ 𝐶 ) ⊆ ( 𝐴 ∨ℋ 𝐶 )
13	9 12	ssini	⊢ ( 𝐵 ∩ 𝐶 ) ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) )
14	2 3	chincli	⊢ ( 𝐵 ∩ 𝐶 ) ∈ Cℋ
15	1 2	chjcli	⊢ ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ
16	1 3	chjcli	⊢ ( 𝐴 ∨ℋ 𝐶 ) ∈ Cℋ
17	15 16	chincli	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) ) ∈ Cℋ
18	1 14 17	chlubi	⊢ ( ( 𝐴 ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) ) ∧ ( 𝐵 ∩ 𝐶 ) ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) ) ) ↔ ( 𝐴 ∨ℋ ( 𝐵 ∩ 𝐶 ) ) ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) ) )
19	18	bicomi	⊢ ( ( 𝐴 ∨ℋ ( 𝐵 ∩ 𝐶 ) ) ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) ) ↔ ( 𝐴 ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) ) ∧ ( 𝐵 ∩ 𝐶 ) ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) ) ) )
20	6 13 19	mpbir2an	⊢ ( 𝐴 ∨ℋ ( 𝐵 ∩ 𝐶 ) ) ⊆ ( ( 𝐴 ∨ℋ 𝐵 ) ∩ ( 𝐴 ∨ℋ 𝐶 ) )