Theorem dmdsl3

Description: Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of MaedaMaeda p. 2. (Contributed by NM, 26-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	dmdsl3	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		dmdi	⊢ ( ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
2	1	exp32	⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐵 𝑀ℋ* 𝐴 → ( 𝐴 ⊆ 𝐶 → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) ) ) )
3	2	3com12	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐵 𝑀ℋ* 𝐴 → ( 𝐴 ⊆ 𝐶 → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) ) ) )
4	3	imp32	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
5	4	3adantr3	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
6		chjcom	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )
7	6	ineq2d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
8	7	3adant3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) )
9		df-ss	⊢ ( 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ↔ ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = 𝐶 )
10	9	biimpi	⊢ ( 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) → ( 𝐶 ∩ ( 𝐴 ∨ℋ 𝐵 ) ) = 𝐶 )
11	8 10	sylan9req	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) → ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) = 𝐶 )
12	11	3ad2antr3	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( 𝐶 ∩ ( 𝐵 ∨ℋ 𝐴 ) ) = 𝐶 )
13	5 12	eqtrd	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐵 𝑀ℋ* 𝐴 ∧ 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) ) → ( ( 𝐶 ∩ 𝐵 ) ∨ℋ 𝐴 ) = 𝐶 )