Metamath Proof Explorer


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 ( ( 𝐴C𝐵C ∧ ( 𝐴𝐵 ) ⋖ 𝐵 ) → 𝐴 𝑀 𝐵 )

Proof

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