Theorem shscl

Description: Closure of subspace sum. (Contributed by NM, 15-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shscl	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		oveq1	$$⊢A=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right)\to A{+}_{\mathscr{H}}B=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}B$$
2	1	eleq1d	$$⊢A=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(A{+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}\leftrightarrow if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}\right)$$
3		oveq2	$$⊢B=if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\to if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}B=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)$$
4	3	eleq1d	$$⊢B=if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}\leftrightarrow if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{S}}_{\mathscr{H}}\right)$$
5		helsh	$$⊢\mathscr{H}\in {\mathbf{S}}_{\mathscr{H}}$$
6	5	elimel	$$⊢if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right)\in {\mathbf{S}}_{\mathscr{H}}$$
7	5	elimel	$$⊢if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{S}}_{\mathscr{H}}$$
8	6 7	shscli	$$⊢if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{S}}_{\mathscr{H}}$$
9	2 4 8	dedth2h	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B\in {\mathbf{S}}_{\mathscr{H}}$$