Theorem chjjdiri

Description: Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chj12.1	⊢ 𝐴 ∈ Cℋ
	chj12.2	⊢ 𝐵 ∈ Cℋ
	chj12.3	⊢ 𝐶 ∈ Cℋ
Assertion	chjjdiri	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( ( 𝐴 ∨ℋ 𝐶 ) ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		chj12.1	⊢ 𝐴 ∈ Cℋ
2		chj12.2	⊢ 𝐵 ∈ Cℋ
3		chj12.3	⊢ 𝐶 ∈ Cℋ
4	3	chjidmi	⊢ ( 𝐶 ∨ℋ 𝐶 ) = 𝐶
5	4	oveq2i	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ ( 𝐶 ∨ℋ 𝐶 ) ) = ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 )
6	1 2 3 3	chj4i	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ ( 𝐶 ∨ℋ 𝐶 ) ) = ( ( 𝐴 ∨ℋ 𝐶 ) ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) )
7	5 6	eqtr3i	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( ( 𝐴 ∨ℋ 𝐶 ) ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) )