Theorem shslej

Description: Subspace sum is smaller than subspace join. Remark in Kalmbach p. 65. (Contributed by NM, 12-Jul-2004) (New usage is discouraged.)

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

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		oveq1	$$⊢A=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right)\to A{\vee }_{\mathscr{H}}B=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B$$
3	1 2	sseq12d	$$⊢A=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(A{+}_{\mathscr{H}}B\subseteq A{\vee }_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}B\subseteq if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B\right)$$
4		oveq2	$$⊢B=if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\to if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}B=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)$$
5		oveq2	$$⊢B=if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\to if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B=if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)$$
6	4 5	sseq12d	$$⊢B=if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}B\subseteq if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\subseteq if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\right)$$
7		helsh	$$⊢\mathscr{H}\in {\mathbf{S}}_{\mathscr{H}}$$
8	7	elimel	$$⊢if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right)\in {\mathbf{S}}_{\mathscr{H}}$$
9	7	elimel	$$⊢if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{S}}_{\mathscr{H}}$$
10	8 9	shsleji	$$⊢if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){+}_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)\subseteq if\left(A\in {\mathbf{S}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{S}}_{\mathscr{H}},B,\mathscr{H}\right)$$
11	3 6 10	dedth2h	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B\subseteq A{\vee }_{\mathscr{H}}B$$