Theorem qlax4i

Description: One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-4" 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	qlax4i	⊢ ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐵 ∨ℋ ( ⊥ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		qlax.1	⊢ 𝐴 ∈ Cℋ
2		qlax.2	⊢ 𝐵 ∈ Cℋ
3	1	chj1i	⊢ ( 𝐴 ∨ℋ ℋ ) = ℋ
4	2	chjoi	⊢ ( 𝐵 ∨ℋ ( ⊥ ‘ 𝐵 ) ) = ℋ
5	4	oveq2i	⊢ ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∨ℋ ℋ )
6	3 5 4	3eqtr4i	⊢ ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐵 ∨ℋ ( ⊥ ‘ 𝐵 ) )