This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )