Theorem shsvs

Description: Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shsvs	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 −ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		shscl	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 +ℋ 𝐵 ) ∈ Sℋ )
2	1	a1d	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐴 +ℋ 𝐵 ) ∈ Sℋ ) )
3		shsel1	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐶 ∈ 𝐴 → 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
4	3	adantrd	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
5		shsel2	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐷 ∈ 𝐵 → 𝐷 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
6	5	adantld	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐷 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
7	2 4 6	3jcad	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐴 +ℋ 𝐵 ) ∈ Sℋ ∧ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 +ℋ 𝐵 ) ) ) )
8		shsubcl	⊢ ( ( ( 𝐴 +ℋ 𝐵 ) ∈ Sℋ ∧ 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 +ℋ 𝐵 ) ) → ( 𝐶 −ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) )
9	7 8	syl6	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 −ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) ) )