Theorem cvmd

Description: The covering property implies the modular pair property. Lemma 7.5.1 of MaedaMaeda p. 31. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvmd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ) → 𝐴 𝑀ℋ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ineq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∩ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) )
2	1	breq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ⋖ℋ 𝐵 ) )
3		breq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ 𝐵 ) )
4	2 3	imbi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵 ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ⋖ℋ 𝐵 → if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ 𝐵 ) ) )
5		ineq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
6		id	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) )
7	5 6	breq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⋖ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
8		breq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
9	7 8	imbi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ⋖ℋ 𝐵 → if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ 𝐵 ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⋖ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
10		ifchhv	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
11		ifchhv	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
12	10 11	cvmdi	⊢ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ⋖ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) 𝑀ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) )
13	4 9 12	dedth2h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵 ) )
14	13	3impia	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ) → 𝐴 𝑀ℋ 𝐵 )