Theorem pjoml3

Description: Variation of orthomodular law. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	pjoml3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sseq2	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
2		id	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) )
3		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
4	3	oveq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) )
5	2 4	ineq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) )
6	5	eqeq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = 𝐵 ) )
7	1 6	imbi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ) ↔ ( 𝐵 ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = 𝐵 ) ) )
8		sseq1	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( 𝐵 ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
9		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
10	9	ineq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
11		id	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) )
12	10 11	eqeq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
13	8 12	imbi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( 𝐵 ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ 𝐵 ) ) = 𝐵 ) ↔ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
14		ifchhv	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
15		ifchhv	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
16	14 15	pjoml3i	⊢ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) )
17	7 13 16	dedth2h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = 𝐵 ) )