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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 ( ( 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 )