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Metamath Proof Explorer

Theorem cvmdi

Description: The covering property implies the modular pair property. Lemma 7.5.1 of MaedaMaeda p. 31. (Contributed by NM, 16-Jun-2004) (New usage is discouraged.)

Ref Expression
Hypotheses mdsl.1 ⊢ 𝐴 ∈ Cℋ
mdsl.2 ⊢ 𝐵 ∈ Cℋ
Assertion cvmdi ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵 )

Proof

Step Hyp Ref Expression
1 mdsl.1 ⊢ 𝐴 ∈ Cℋ
2 mdsl.2 ⊢ 𝐵 ∈ Cℋ
3 anass ⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ∧ 𝑥 ⊆ 𝐵 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ ( 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ∧ 𝑥 ⊆ 𝐵 ) ) )
4 2 1 chub2i ⊢ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 )
5 sstr ⊢ ( ( 𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) )
6 4 5 mpan2 ⊢ ( 𝑥 ⊆ 𝐵 → 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) )
7 6 pm4.71ri ⊢ ( 𝑥 ⊆ 𝐵 ↔ ( 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ∧ 𝑥 ⊆ 𝐵 ) )
8 7 anbi2i ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ ( 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ∧ 𝑥 ⊆ 𝐵 ) ) )
9 3 8 bitr4i ⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ∧ 𝑥 ⊆ 𝐵 ) ↔ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) )
10 1 2 chincli ⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ
11 cvnbtwn4 ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( 𝑥 = ( 𝐴 ∩ 𝐵 ) ∨ 𝑥 = 𝐵 ) ) ) )
12 10 2 11 mp3an12 ⊢ ( 𝑥 ∈ Cℋ → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( 𝑥 = ( 𝐴 ∩ 𝐵 ) ∨ 𝑥 = 𝐵 ) ) ) )
13 12 impcom ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ∧ 𝑥 ∈ Cℋ ) → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( 𝑥 = ( 𝐴 ∩ 𝐵 ) ∨ 𝑥 = 𝐵 ) ) )
14 10 1 chjcomi ⊢ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) )
15 1 2 chabs1i ⊢ ( 𝐴 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = 𝐴
16 14 15 eqtri ⊢ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) = 𝐴
17 16 ineq1i ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐴 ∩ 𝐵 )
18 10 chjidmi ⊢ ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∩ 𝐵 )
19 17 18 eqtr4i ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) )
20 oveq1 ⊢ ( 𝑥 = ( 𝐴 ∩ 𝐵 ) → ( 𝑥 ∨ℋ 𝐴 ) = ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) )
21 20 ineq1d ⊢ ( 𝑥 = ( 𝐴 ∩ 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( ( 𝐴 ∩ 𝐵 ) ∨ℋ 𝐴 ) ∩ 𝐵 ) )
22 oveq1 ⊢ ( 𝑥 = ( 𝐴 ∩ 𝐵 ) → ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝐴 ∩ 𝐵 ) ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )
23 19 21 22 3eqtr4a ⊢ ( 𝑥 = ( 𝐴 ∩ 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )
24 incom ⊢ ( ( 𝐵 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐵 ∩ ( 𝐵 ∨ℋ 𝐴 ) )
25 2 1 chabs2i ⊢ ( 𝐵 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) = 𝐵
26 2 1 chabs1i ⊢ ( 𝐵 ∨ℋ ( 𝐵 ∩ 𝐴 ) ) = 𝐵
27 incom ⊢ ( 𝐵 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐵 )
28 27 oveq2i ⊢ ( 𝐵 ∨ℋ ( 𝐵 ∩ 𝐴 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∩ 𝐵 ) )
29 25 26 28 3eqtr2i ⊢ ( 𝐵 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∩ 𝐵 ) )
30 24 29 eqtri ⊢ ( ( 𝐵 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐵 ∨ℋ ( 𝐴 ∩ 𝐵 ) )
31 oveq1 ⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∨ℋ 𝐴 ) = ( 𝐵 ∨ℋ 𝐴 ) )
32 31 ineq1d ⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝐵 ∨ℋ 𝐴 ) ∩ 𝐵 ) )
33 oveq1 ⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )
34 30 32 33 3eqtr4a ⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )
35 23 34 jaoi ⊢ ( ( 𝑥 = ( 𝐴 ∩ 𝐵 ) ∨ 𝑥 = 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )
36 13 35 syl6 ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ∧ 𝑥 ∈ Cℋ ) → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) )
37 9 36 biimtrid ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ∧ 𝑥 ∈ Cℋ ) → ( ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ∧ 𝑥 ⊆ 𝐵 ) → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) )
38 37 exp4b ⊢ ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → ( 𝑥 ∈ Cℋ → ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ) )
39 38 ralrimiv ⊢ ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → ∀ 𝑥 ∈ Cℋ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) )
40 1 2 mdsl1i ⊢ ( ∀ 𝑥 ∈ Cℋ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) ↔ 𝐴 𝑀ℋ 𝐵 )
41 39 40 sylib ⊢ ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵 )