Theorem cmmdi

Description: Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sumdmdi.1	⊢ 𝐴 ∈ Cℋ
2		sumdmdi.2	⊢ 𝐵 ∈ Cℋ
3	1 2	cmcm4i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ ( ⊥ ‘ 𝐵 ) )
4	1	choccli	⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ
5	2	choccli	⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ
6	4 5	osumcor2i	⊢ ( ( ⊥ ‘ 𝐴 ) 𝐶ℋ ( ⊥ ‘ 𝐵 ) → ( ( ⊥ ‘ 𝐴 ) +ℋ ( ⊥ ‘ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) )
7	3 6	sylbi	⊢ ( 𝐴 𝐶ℋ 𝐵 → ( ( ⊥ ‘ 𝐴 ) +ℋ ( ⊥ ‘ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) )
8	4 5	sumdmdii	⊢ ( ( ( ⊥ ‘ 𝐴 ) +ℋ ( ⊥ ‘ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) → ( ⊥ ‘ 𝐴 ) 𝑀ℋ* ( ⊥ ‘ 𝐵 ) )
9	7 8	syl	⊢ ( 𝐴 𝐶ℋ 𝐵 → ( ⊥ ‘ 𝐴 ) 𝑀ℋ* ( ⊥ ‘ 𝐵 ) )
10		mddmd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ* ( ⊥ ‘ 𝐵 ) ) )
11	1 2 10	mp2an	⊢ ( 𝐴 𝑀ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ* ( ⊥ ‘ 𝐵 ) )
12	9 11	sylibr	⊢ ( 𝐴 𝐶ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵 )