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Metamath Proof Explorer

Theorem mdslmd4i

Description: Modular pair condition that implies the modular pair property in a sublattice. Lemma 1.5.2 of MaedaMaeda p. 2. (Contributed by NM, 24-Dec-2006) (New usage is discouraged.)

Ref Expression
Hypotheses mdslmd.1 ⊢ 𝐴 ∈ Cℋ
mdslmd.2 ⊢ 𝐵 ∈ Cℋ
mdslmd.3 ⊢ 𝐶 ∈ Cℋ
mdslmd.4 ⊢ 𝐷 ∈ Cℋ
Assertion mdslmd4i ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → 𝐶 𝑀ℋ 𝐷 )

Proof

Step Hyp Ref Expression
1 mdslmd.1 ⊢ 𝐴 ∈ Cℋ
2 mdslmd.2 ⊢ 𝐵 ∈ Cℋ
3 mdslmd.3 ⊢ 𝐶 ∈ Cℋ
4 mdslmd.4 ⊢ 𝐷 ∈ Cℋ
5 simp1 ⊢ ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → 𝐴 𝑀ℋ 𝐵 )
6 1 2 chincli ⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ
7 ssmd1 ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ∧ 𝐷 ∈ Cℋ ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ) → ( 𝐴 ∩ 𝐵 ) 𝑀ℋ 𝐷 )
8 6 4 7 mp3an12 ⊢ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 → ( 𝐴 ∩ 𝐵 ) 𝑀ℋ 𝐷 )
9 8 adantr ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) → ( 𝐴 ∩ 𝐵 ) 𝑀ℋ 𝐷 )
10 9 3ad2ant3 ⊢ ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → ( 𝐴 ∩ 𝐵 ) 𝑀ℋ 𝐷 )
11 sslin ⊢ ( 𝐷 ⊆ 𝐵 → ( 𝐴 ∩ 𝐷 ) ⊆ ( 𝐴 ∩ 𝐵 ) )
12 sstr ⊢ ( ( ( 𝐴 ∩ 𝐷 ) ⊆ ( 𝐴 ∩ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ) → ( 𝐴 ∩ 𝐷 ) ⊆ 𝐶 )
13 11 12 sylan ⊢ ( ( 𝐷 ⊆ 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ) → ( 𝐴 ∩ 𝐷 ) ⊆ 𝐶 )
14 13 ancoms ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐷 ⊆ 𝐵 ) → ( 𝐴 ∩ 𝐷 ) ⊆ 𝐶 )
15 14 ad2ant2rl ⊢ ( ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → ( 𝐴 ∩ 𝐷 ) ⊆ 𝐶 )
16 15 3adant1 ⊢ ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → ( 𝐴 ∩ 𝐷 ) ⊆ 𝐶 )
17 simp2r ⊢ ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → 𝐶 ⊆ 𝐴 )
18 1 2 4 3 mdslmd3i ⊢ ( ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) 𝑀ℋ 𝐷 ) ∧ ( ( 𝐴 ∩ 𝐷 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ) → 𝐶 𝑀ℋ ( 𝐵 ∩ 𝐷 ) )
19 5 10 16 17 18 syl22anc ⊢ ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → 𝐶 𝑀ℋ ( 𝐵 ∩ 𝐷 ) )
20 sseqin2 ⊢ ( 𝐷 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐷 ) = 𝐷 )
21 20 biimpi ⊢ ( 𝐷 ⊆ 𝐵 → ( 𝐵 ∩ 𝐷 ) = 𝐷 )
22 21 adantl ⊢ ( ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) → ( 𝐵 ∩ 𝐷 ) = 𝐷 )
23 22 3ad2ant3 ⊢ ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → ( 𝐵 ∩ 𝐷 ) = 𝐷 )
24 19 23 breqtrd ⊢ ( ( 𝐴 𝑀ℋ 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) → 𝐶 𝑀ℋ 𝐷 )