Theorem shlej2

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

	Ref	Expression
Assertion	shlej2	⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		shlej1	⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )
2		shjcom	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐶 ) = ( 𝐶 ∨ℋ 𝐴 ) )
3	2	3adant2	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐶 ) = ( 𝐶 ∨ℋ 𝐴 ) )
4	3	adantr	⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ 𝐶 ) = ( 𝐶 ∨ℋ 𝐴 ) )
5		shjcom	⊢ ( ( 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → ( 𝐵 ∨ℋ 𝐶 ) = ( 𝐶 ∨ℋ 𝐵 ) )
6	5	3adant1	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → ( 𝐵 ∨ℋ 𝐶 ) = ( 𝐶 ∨ℋ 𝐵 ) )
7	6	adantr	⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐵 ∨ℋ 𝐶 ) = ( 𝐶 ∨ℋ 𝐵 ) )
8	1 4 7	3sstr3d	⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )