Theorem shsval

Description: Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in Kalmbach p. 65. (Contributed by NM, 16-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	shsval	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B={+}_{ℎ}\left[A\times B\right]$$

Proof

Step	Hyp	Ref	Expression
1		xpeq12	$$⊢\left(x=A\wedge y=B\right)\to x\times y=A\times B$$
2	1	imaeq2d	$$⊢\left(x=A\wedge y=B\right)\to {+}_{ℎ}\left[x\times y\right]={+}_{ℎ}\left[A\times B\right]$$
3		df-shs	$$⊢{+}_{\mathscr{H}}=\left(x\in {\mathbf{S}}_{\mathscr{H}},y\in {\mathbf{S}}_{\mathscr{H}}⟼{+}_{ℎ}\left[x\times y\right]\right)$$
4		hilablo	$$⊢{+}_{ℎ}\in \mathrm{AbelOp}$$
5		imaexg	$$⊢{+}_{ℎ}\in \mathrm{AbelOp}\to {+}_{ℎ}\left[A\times B\right]\in \mathrm{V}$$
6	4 5	ax-mp	$$⊢{+}_{ℎ}\left[A\times B\right]\in \mathrm{V}$$
7	2 3 6	ovmpoa	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B={+}_{ℎ}\left[A\times B\right]$$