Theorem shsvs

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

	Ref	Expression
Assertion	shsvs	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to C{-}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		shscl	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}$$
2	1	a1d	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to A{+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}\right)$$
3		shsel1	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(C\in A\to C\in \left(A{+}_{\mathscr{H}}B\right)\right)$$
4	3	adantrd	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to C\in \left(A{+}_{\mathscr{H}}B\right)\right)$$
5		shsel2	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(D\in B\to D\in \left(A{+}_{\mathscr{H}}B\right)\right)$$
6	5	adantld	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to D\in \left(A{+}_{\mathscr{H}}B\right)\right)$$
7	2 4 6	3jcad	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to \left(A{+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}\wedge C\in \left(A{+}_{\mathscr{H}}B\right)\wedge D\in \left(A{+}_{\mathscr{H}}B\right)\right)\right)$$
8		shsubcl	$$⊢\left(A{+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}\wedge C\in \left(A{+}_{\mathscr{H}}B\right)\wedge D\in \left(A{+}_{\mathscr{H}}B\right)\right)\to C{-}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)$$
9	7 8	syl6	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to C{-}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)\right)$$