Theorem cmdmdi

Description: Commuting subspaces form a dual modular pair. (Contributed by NM, 25-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	sumdmdi.1	⊢ 𝐴 ∈ Cℋ
	sumdmdi.2	⊢ 𝐵 ∈ Cℋ
Assertion	cmdmdi	⊢ ( 𝐴 𝐶ℋ 𝐵 → 𝐴 𝑀ℋ* 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sumdmdi.1	⊢ 𝐴 ∈ Cℋ
2		sumdmdi.2	⊢ 𝐵 ∈ Cℋ
3	1	choccli	⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ
4	2	choccli	⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ
5	3 4	cmmdi	⊢ ( ( ⊥ ‘ 𝐴 ) 𝐶ℋ ( ⊥ ‘ 𝐵 ) → ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) )
6	1 2	cmcm4i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ ( ⊥ ‘ 𝐵 ) )
7		dmdmd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ) )
8	1 2 7	mp2an	⊢ ( 𝐴 𝑀ℋ* 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) )
9	5 6 8	3imtr4i	⊢ ( 𝐴 𝐶ℋ 𝐵 → 𝐴 𝑀ℋ* 𝐵 )