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

Metamath Proof Explorer

Theorem chjval

Description: Value of join in CH . (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

Ref Expression
Assertion chjval ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 chsh ⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2 chsh ⊢ ( 𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ )
3 shjval ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )