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 ) ) )