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

Metamath Proof Explorer

Theorem chj12

Description: A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion chj12 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 chjcom ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )
2 1 3adant3 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )
3 2 oveq1d ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( ( 𝐵 ∨ℋ 𝐴 ) ∨ℋ 𝐶 ) )
4 chjass ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) )
5 chjass ⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐵 ∨ℋ 𝐴 ) ∨ℋ 𝐶 ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) ) )
6 5 3com12 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐵 ∨ℋ 𝐴 ) ∨ℋ 𝐶 ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) ) )
7 3 4 6 3eqtr3d ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) ) )