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Metamath Proof Explorer

Theorem pjomli

Description: Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of Kalmbach p. 22. Derived using projections; compare omlsi . (Contributed by NM, 6-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml.1	⊢ 𝐴 ∈ C_ℋ
	pjoml.2	⊢ 𝐵 ∈ S_ℋ
Assertion	pjomli	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ ( 𝐵 ∩ ( ⊥ ‘ 𝐴 ) ) = 0_ℋ ) → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		pjoml.1	⊢ 𝐴 ∈ C_ℋ
2		pjoml.2	⊢ 𝐵 ∈ S_ℋ
3	1 2	omlsi	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ ( 𝐵 ∩ ( ⊥ ‘ 𝐴 ) ) = 0_ℋ ) → 𝐴 = 𝐵 )