Theorem shlej2i

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

	Ref	Expression
Hypotheses	shincl.1	⊢ 𝐴 ∈ Sℋ
	shincl.2	⊢ 𝐵 ∈ Sℋ
	shless.1	⊢ 𝐶 ∈ Sℋ
Assertion	shlej2i	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	⊢ 𝐴 ∈ Sℋ
2		shincl.2	⊢ 𝐵 ∈ Sℋ
3		shless.1	⊢ 𝐶 ∈ Sℋ
4	1 2 3	shlej1i	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )
5	3 1	shjcomi	⊢ ( 𝐶 ∨ℋ 𝐴 ) = ( 𝐴 ∨ℋ 𝐶 )
6	3 2	shjcomi	⊢ ( 𝐶 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐶 )
7	4 5 6	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )