Theorem pjoml5i

Description: The orthomodular law. Remark in Kalmbach p. 22. (Contributed by NM, 12-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
Assertion	pjoml5i	⊢ ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) = ( 𝐴 ∨ℋ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3	1 2	chub1i	⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 )
4	1 2	chjcli	⊢ ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ
5	1 4	pjoml2i	⊢ ( 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) = ( 𝐴 ∨ℋ 𝐵 ) )
6	3 5	ax-mp	⊢ ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ ( 𝐴 ∨ℋ 𝐵 ) ) ) = ( 𝐴 ∨ℋ 𝐵 )