Theorem qlax5i

Description: One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	qlax.1	⊢ 𝐴 ∈ Cℋ
	qlax.2	⊢ 𝐵 ∈ Cℋ
Assertion	qlax5i	⊢ ( 𝐴 ∨ℋ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		qlax.1	⊢ 𝐴 ∈ Cℋ
2		qlax.2	⊢ 𝐵 ∈ Cℋ
3	1 2	chdmj2i	⊢ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) )
4	3	oveq2i	⊢ ( 𝐴 ∨ℋ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) ) = ( 𝐴 ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) )
5	2	choccli	⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ
6	1 5	chabs1i	⊢ ( 𝐴 ∨ℋ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) = 𝐴
7	4 6	eqtri	⊢ ( 𝐴 ∨ℋ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) ) = 𝐴