Theorem pjoml2i

Description: Variation of orthomodular law. Definition in Kalmbach p. 22. (Contributed by NM, 31-Oct-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
Assertion	pjoml2i	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3		inss2	⊢ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ⊆ 𝐵
4	1	choccli	⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ
5	4 2	chincli	⊢ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ∈ Cℋ
6	1 5 2	chlubii	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ⊆ 𝐵 ) → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ⊆ 𝐵 )
7	3 6	mpan2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ⊆ 𝐵 )
8	1 5	chdmj1i	⊢ ( ⊥ ‘ ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) )
9	8	ineq2i	⊢ ( 𝐵 ∩ ( ⊥ ‘ ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) ) = ( 𝐵 ∩ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) )
10		incom	⊢ ( 𝐵 ∩ ( ⊥ ‘ 𝐴 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 )
11	10	ineq1i	⊢ ( ( 𝐵 ∩ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) = ( ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ∩ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) )
12		inass	⊢ ( ( 𝐵 ∩ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) = ( 𝐵 ∩ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) )
13	5	chocini	⊢ ( ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ∩ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) = 0ℋ
14	11 12 13	3eqtr3i	⊢ ( 𝐵 ∩ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) ) = 0ℋ
15	9 14	eqtri	⊢ ( 𝐵 ∩ ( ⊥ ‘ ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) ) = 0ℋ
16	1 5	chjcli	⊢ ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ∈ Cℋ
17	2	chshii	⊢ 𝐵 ∈ Sℋ
18	16 17	pjomli	⊢ ( ( ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ⊆ 𝐵 ∧ ( 𝐵 ∩ ( ⊥ ‘ ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) ) ) = 0ℋ ) → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = 𝐵 )
19	7 15 18	sylancl	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = 𝐵 )