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 )