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Metamath Proof Explorer

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 )