Theorem shsubcl

Description: Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	shsubcl	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻 ) → ( 𝐴 −ℎ 𝐵 ) ∈ 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		shss	⊢ ( 𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ )
2	1	sseld	⊢ ( 𝐻 ∈ Sℋ → ( 𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ ) )
3	1	sseld	⊢ ( 𝐻 ∈ Sℋ → ( 𝐵 ∈ 𝐻 → 𝐵 ∈ ℋ ) )
4	2 3	anim12d	⊢ ( 𝐻 ∈ Sℋ → ( ( 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻 ) → ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ) )
5	4	3impib	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻 ) → ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) )
6		hvsubval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 −ℎ 𝐵 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) )
7	5 6	syl	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻 ) → ( 𝐴 −ℎ 𝐵 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) )
8		neg1cn	⊢ - 1 ∈ ℂ
9		shmulcl	⊢ ( ( 𝐻 ∈ Sℋ ∧ - 1 ∈ ℂ ∧ 𝐵 ∈ 𝐻 ) → ( - 1 ·ℎ 𝐵 ) ∈ 𝐻 )
10	8 9	mp3an2	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐵 ∈ 𝐻 ) → ( - 1 ·ℎ 𝐵 ) ∈ 𝐻 )
11	10	3adant2	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻 ) → ( - 1 ·ℎ 𝐵 ) ∈ 𝐻 )
12		shaddcl	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ ( - 1 ·ℎ 𝐵 ) ∈ 𝐻 ) → ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) ∈ 𝐻 )
13	11 12	syld3an3	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻 ) → ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) ∈ 𝐻 )
14	7 13	eqeltrd	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻 ) → ( 𝐴 −ℎ 𝐵 ) ∈ 𝐻 )