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Metamath Proof Explorer

Theorem chlej1

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

Ref Expression
Assertion chlej1 ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 chsh ⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2 chsh ⊢ ( 𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ )
3 chsh ⊢ ( 𝐶 ∈ Cℋ → 𝐶 ∈ Sℋ )
4 shlej1 ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )
5 1 2 3 4 syl3anl ⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )